Billy and Tommy are brothers, aged 6 and 7. Minutes ago, they had been playing cordially together with their toys. But then Billy accidentally broke one of Tommy's toys, prompting Tommy to call Billy a nasty name. The fight quickly escalated, but the boys did not physically strike each other, since they always got in trouble with Mom when they did. So now they are standing in the living room, staring at each other, fire blazing in their eyes.
Mom is talking on the phone with a friend of hers on the couch. She keeps an eye on her sons, but is simply carrying on with her conversation. The boys periodically glance back at her, each hoping that she will intervene and side with him. They take turns mouthing nasty names at each other.
Tommy calls Billy an especially nasty name. Suddenly, Billy gets so frustrated that he cannot stop himself. He slaps his brother across the arm. Tommy immediately points to his brother, looks at Mom, and says to her, "See!? See!? He hit me first!"
Mom does not intervene yet. So Tommy hits Billy back. The two continue hitting each other until Mom says to the person on the other line that she'll have to call them back. Mom intervenes and physically separates the two boys. She punishes both. "But he started it! He hit me first!" protests Tommy.
* * *
Two opposing groups, white supremacists and Antifa, are both in a park, hurling insults at each other, fire blazing in their eyes.
The police are standing along the edge of the park. They keep an eye on the protestors, but are simply doing their job to maintain order. The protestors periodically glance back at them, each side hoping that the police will intervene and side with them. The sides continue provoking each other.
Camera crews are scattered throughout the park. They keep an eye on the apparent leaders of each faction, but are simply covering the event. The protestors periodically glance back at them, each side hoping that reporters will give them a platform to broadcast their denouncement of the other side. The sides continue provoking each other.
Someone on one side says something that causes someone on the other side to punch.
In the days that follow, members and supporters of each side assert that the other side hit first, and that they were only defending themselves.
* * *
Both sides want to convey the narrative that "the other side" is the violent one--not them. So it matters who hit first, because after the first blow, all subsequent violence is apparently irrelevant. Each side wants to gain outsiders' sympathy by claiming they were the victim of the first blow. It doesn't fit their narrative if both sides are violent.
Saturday, November 11, 2017
Friday, October 27, 2017
Blade Runner 2049
I saw the new Blade Runner movie. The production was great; the plot was filled with holes. Below are the questions that I had after watching the film. These questions contain spoilers, so don't read this if you haven't seen the film yet and care about the plot not being spoiled.
Characters:
K (Ryan Gosling)
Rick Deckard (Harrison Ford)
Joi (Ana de Armas)
Luv (Sylvia Hoeks)
Lt. Joshi (Robin Wright)
Dr. Ana Stelline (Carla Juri)
Niander Wallace (Jared Leto)
Freysa (Hiam Abbass)
Mariette (Mackenzie Davis)
Questions:
1) How did K deduce that Dr. Stelline is Deckard's daughter? I think the viewer is supposed to make this connection by the fact that Dr. Stelline got emotional about K's memory of the horse when she confirmed its authenticity. But that is an incredibly weak connection.
2) Why was Luv trying to kill K? Perhaps she was only indirectly trying to kill him, as K impeded her in her pursuit of Deckard. But then why did her team attempt to blow up the old casino that K and Deckard were in, especially if she wanted Deckard alive?
3) Why did Wallace and K not work together? Both characters had the same goal--to find Deckard, although their motivations for doing so were different.
4) What happened with Wallace? Wallace did not have any additional screen time after his failed attempt at extracting information from Deckard. So as far as we know, he is still out there, attempting to accomplish his goal. This was a loose end that was not tied up.
5) What happened with Freysa and Mariette? Neither of these characters had any additional screen time after they rescued K and told him of the replicant resistance movement. They did not provide K with any assistance. So what purpose did that scene--really, even the existence of those characters--serve? This was a loose end that was not tied up.
6) Why did Deckard not just delete the DNA record of his daughter if he was trying to keep her hidden? Deckard clearly had not just access to the central database of DNA records, but also editing rights, as he was able to scramble his daughter's birth record and list her as dead. Why did he not just delete the record?
7) Why should K think that he was the child born of the replicant mother, Rachael? K had the memory of the toy horse with the inscribed date on it, which matched the date inscribed on the tree. This was how he theorized that he was the child of Rachael, I think. But K also knew that all his memories were implants. Why would the memory of this toy horse make him think otherwise?
8) Why did Mariette place the tracking device in K's coat? Why was K special such that she needed to track him? If it was to convince him to join the replicant resistance movement, does that mean that she tracked other replicants?
Related: Did Joi or K summon Mariette to simulate sex with K?
9) Why did K go to Las Vegas? Easy answer is because the wood that K's toy horse was made from was traced to Las Vegas. But so what? The wood could have been imported from anywhere. Did he really think that he would find his supposed father, who turned out to be Deckard, there? That is incredibly shaky reasoning. (But, fortunately, due to poor plot writing, K was able to find exactly whom he was looking for in the first building he entered. How convenient.)
10) Who stole the orphanage's records? What motivation did the thief have to steal these records? What information was in these records that was so valuable? This plot line was not at all pursued, and I suspect it might have been more important and helpful for K's search.
11) Why would K have the physical toy horse in his possession if the memory of the toy horse was not actually his?
12) What constitutes an acceptable replicant obedience test? As far as I could tell, the only difference between the first and second obedience tests administered to K at the LAPD office was the promptness of K's responses. Why would that force him to retire in 48 hours, as Lt. Joshi told him?
Related: How did that test ensure replicant obedience? The test seemed to require the replicant to say one-word responses, usually the last word that the reader said. Anybody could pass that test to give the illusion of compliance, and then immediately rebel.
13) Why did K have to run away after failing his obedience test? Who was coming after him? Why?
14) Why does Freysa have knowledge about Deckard's child? Naturally, she would have a vested interest in Deckard's child--a child of a replicant--but how did Freysa get access to that information? Was she in contact with Deckard? Does she have a connection to Deckard?
15) How did Luv's team know that Deckard would be with K in Las Vegas? Luv knew K was out trying to find the father of the replicant child (it's why she killed Lt. Joshi), but how did she know that K had found him when she pulled up his geolocation information on Joshi's computer?
Characters:
K (Ryan Gosling)
Rick Deckard (Harrison Ford)
Joi (Ana de Armas)
Luv (Sylvia Hoeks)
Lt. Joshi (Robin Wright)
Dr. Ana Stelline (Carla Juri)
Niander Wallace (Jared Leto)
Freysa (Hiam Abbass)
Mariette (Mackenzie Davis)
Questions:
1) How did K deduce that Dr. Stelline is Deckard's daughter? I think the viewer is supposed to make this connection by the fact that Dr. Stelline got emotional about K's memory of the horse when she confirmed its authenticity. But that is an incredibly weak connection.
2) Why was Luv trying to kill K? Perhaps she was only indirectly trying to kill him, as K impeded her in her pursuit of Deckard. But then why did her team attempt to blow up the old casino that K and Deckard were in, especially if she wanted Deckard alive?
3) Why did Wallace and K not work together? Both characters had the same goal--to find Deckard, although their motivations for doing so were different.
4) What happened with Wallace? Wallace did not have any additional screen time after his failed attempt at extracting information from Deckard. So as far as we know, he is still out there, attempting to accomplish his goal. This was a loose end that was not tied up.
5) What happened with Freysa and Mariette? Neither of these characters had any additional screen time after they rescued K and told him of the replicant resistance movement. They did not provide K with any assistance. So what purpose did that scene--really, even the existence of those characters--serve? This was a loose end that was not tied up.
6) Why did Deckard not just delete the DNA record of his daughter if he was trying to keep her hidden? Deckard clearly had not just access to the central database of DNA records, but also editing rights, as he was able to scramble his daughter's birth record and list her as dead. Why did he not just delete the record?
7) Why should K think that he was the child born of the replicant mother, Rachael? K had the memory of the toy horse with the inscribed date on it, which matched the date inscribed on the tree. This was how he theorized that he was the child of Rachael, I think. But K also knew that all his memories were implants. Why would the memory of this toy horse make him think otherwise?
8) Why did Mariette place the tracking device in K's coat? Why was K special such that she needed to track him? If it was to convince him to join the replicant resistance movement, does that mean that she tracked other replicants?
Related: Did Joi or K summon Mariette to simulate sex with K?
9) Why did K go to Las Vegas? Easy answer is because the wood that K's toy horse was made from was traced to Las Vegas. But so what? The wood could have been imported from anywhere. Did he really think that he would find his supposed father, who turned out to be Deckard, there? That is incredibly shaky reasoning. (But, fortunately, due to poor plot writing, K was able to find exactly whom he was looking for in the first building he entered. How convenient.)
10) Who stole the orphanage's records? What motivation did the thief have to steal these records? What information was in these records that was so valuable? This plot line was not at all pursued, and I suspect it might have been more important and helpful for K's search.
11) Why would K have the physical toy horse in his possession if the memory of the toy horse was not actually his?
12) What constitutes an acceptable replicant obedience test? As far as I could tell, the only difference between the first and second obedience tests administered to K at the LAPD office was the promptness of K's responses. Why would that force him to retire in 48 hours, as Lt. Joshi told him?
Related: How did that test ensure replicant obedience? The test seemed to require the replicant to say one-word responses, usually the last word that the reader said. Anybody could pass that test to give the illusion of compliance, and then immediately rebel.
13) Why did K have to run away after failing his obedience test? Who was coming after him? Why?
14) Why does Freysa have knowledge about Deckard's child? Naturally, she would have a vested interest in Deckard's child--a child of a replicant--but how did Freysa get access to that information? Was she in contact with Deckard? Does she have a connection to Deckard?
15) How did Luv's team know that Deckard would be with K in Las Vegas? Luv knew K was out trying to find the father of the replicant child (it's why she killed Lt. Joshi), but how did she know that K had found him when she pulled up his geolocation information on Joshi's computer?
Thursday, October 12, 2017
A high school memoir
One day, the question, "Who is going to be worship leader next school year?" popped into my head.
"Well, hey...I could do it...if I just learned guitar..."
It was September 2009. I had recently begun my junior year at Berean Christian High School. I could play a G, a C, and a D on guitar. I knew how to make a bar chord (saying nothing about my ability to actually do so). And I was at least halfway decent at singing--hell, I was in the top choir at the school.
The idea of being worship leader was so exciting to me that I immediately pulled Grandpa Chet's old Martin guitar--the one that was apparently worth $X,000--out of the closet to start tuning and playing chords. My brother had left a piece of paper somewhere in our room which depicted the finger placement on the frets necessary to make common chords. It'll take a lot of work, I thought, but I have a year. I could do it.
Besides, I thought, even if it doesn't work out, I will have learned how to play guitar. And I could be worship leader for my church youth group.
Berean
The next few months, my mind was occupied with ideas and plans for how I would conduct chapels at school if I were worship leader. I closely observed what Melanie--the figurehead worship leader--and Travis--the actual worship leader--did, and I evaluated which aspects I wanted to keep or throw out. Every time my mind wandered to this topic, an overwhelming feeling of excitement welled up in me.
In November 2009, I scheduled an informal worship night at my church. It was intended to be a trial run for how my worship leading might go. I took great care in planning the set list and practicing for the event. I must have invited thirty or more people. Two showed up. Needless to say, it was disheartening.
But I kept on. I knew one failed event couldn't be what stopped me.
I needed an acoustic-electric guitar. The Martin didn't have a pickup, and there was no way my mom would have let me lug that guitar to and from school for chapel once a week anyway. I went to Guitar Center and tried various models out. I decided I liked best a black Ovation with a curved back. It wasn't too expensive ($300), but it also wasn't the cheapest, so I felt good about that. I added it to my Christmas list. Part of me felt guilty for asking for it since I was still far from an expert at guitar.
My parents bought me the Ovation for Christmas. I liked how it looked and how it felt; however, it had some pitch issues. Adjacent strings could be in pitch with each other at the fifth fret; but when the strings were open, each string got progressively more sharp. A source of the issue was the unusually high action. A Guitar Center employee helped me remove a half-millimeter-thick piece of plastic from under the saddle, which helped, but did not completely fix the issue.
Church
One Saturday in February of 2010, I showed up to my youth group's worship team practice to provide vocals. My goal was twofold: to break the stereotype of female backup singers in youth group bands, and to demonstrate (both to myself and to the group) that I could sing and had no problem doing so in front of the youth group.
Later that month, I accompanied a friend of mine who sang at a funeral at my church. This event not only gave me more practice, but I also viewed it as a marketing effort to my peers at my church that I could be my youth group's worship leader.
Berean
A couple weeks later, on a rainy afternoon after school, I spoke with Melanie and told her about my intention to audition for worship leader. I asked her some questions about her experience in the position, and she asked me about my interest. At the end of the conversation, she told me that a few people had come to her directly and told her that I would make a great worship leader. I was both baffled and encouraged. I had told very few people about my interest in the position; but it was also a good sign that my peers were endorsing me on their own.
It was Tuesday, March 30, 2010. I skipped the lunchtime musical practice to audition for worship leader in front of the student government. In addition to the usual speech, this year's audition also required candidates to lead the student government in a song. I gave a five-minute speech which I had loosely prepared for in which I explained where my desire came from and what I wanted to implement as worship leader. Then I led the student government in "Jesus Paid it All". I left the audition feeling fine. It was nothing spectacular, but as I thought, it was all in God's hands then.
On Friday, April 2, Good Friday, I was supposed to give "Noticias del Día" in Period 1 Spanish IV. I was nervous--not because I had to speak in Spanish in front of fifteen of my classmates, but because they were announcing the results of the student government elections on the intercom with the morning announcements.
"Worship leader is Jonathan Lim."
My heart did not stop pounding, even as the announcements finished and I walked up to the front of the room. My friend Curtis mouthed to me, "I'm sorry." I mouthed back, "It's fine."
At break, Dawn and I were chatting. Apparently she had missed the announcement. She was one of the ones that knew I wanted to go for worship leader. When she heard that Jonathan had won the position, she was indignant. "What!? You could do such a better job!" She was more upset about it than I was.
Melanie found me later in the day. She said it was neither my speech nor my song that didn't get me the position. She still wanted me to be involved with the worship team, so she asked me to audition for the worship team. I declined. I didn't perceive myself as good enough at guitar alone or at singing alone to provide only one of those talents. I could provide both, together, halfway decently, coupled with the leadership aspect, and that was all I was willing to give. I also knew that Mrs. Hazelrigg--the choir director--Melanie, and Travis--the favorite son of the music department--would be involved in deciding who comprised the worship team, and I did not want to be subject to their scrutiny.
The more I thought about it, the more upset I became. Jonathan had never demonstrated any evidence that he had any musical ability. Conversely, I had been in the school's top choir, the Abundant Life Singers, for two years. Historically, the worship leader had never not been a member of ALS, probably because second period (when ALS was scheduled) always followed chapel, so the worship leader could return all the musical equipment to the music room and not miss a more academic class.
But most of my unhappiness came from the fact that I had managed to convince myself that worship leader was what God wanted for me. I thought I had seen so many signs pointing to that path. I began to think that the student government went against what God wanted. Enmity toward the student government, toward Jonathan, and toward Melanie began to build within me.
Church
I took some solace in the thought that I could still lead worship at my youth group.
I had sung in the church's youth group band twice. The first time was when I showed up on my own, and the second time was the following week, when Cory, the worship leader asked me to return. I was never scheduled to return again. Each week, I received an email from Cory with the list of musicians for the following Sunday. And each week, my name was not on the list. It bothered me that I was not able to demonstrate my ability and interest in the worship team; but I also knew that the youth group didn't have much of a choice for the following year's worship leader selection. The current worship band consisted entirely of graduating seniors, with the exception of one terrible vocalist and one electric guitar player, Brent. So I continued refining my skills on guitar, even though I had been rejected for the position at school.
Berean
One evening in June 2010, I had a hard time coping with my Berean worship leader defeat. I wanted a reason. So I reached out to Melanie on Facebook. I asked her what it was that disqualified me, if it was neither my speech nor my song. Her response, saturated with agitation, was that it was nothing personal against me; that student government collectively decided that Jonathan would be a better fit; and that I should talk to the student government adviser, because it was not her place to say.
Less than a week later, she deleted me as a Facebook friend.
Church
One Sunday in early June 2010, Cory was out of town. This is my chance to step up, I thought. So I offered to lead the youth group worship that Sunday. Cory left behind no instructions, no guidance, no access to sheet music. So of course none of the electrical equipment worked that morning, and I was left scrambling to figure out what to do in front of my peers and my youth pastor. It felt as though I had been deliberately set up for failure.
Brent and I got together another Saturday that summer to practice for Sunday's worship. As we were leaving his house, his mom called out, suggesting that I lead worship for the youth group during the school year. She said it seemed logical, as I was graduating and Brent was only an incoming sophomore. I admitted that I had been hoping to step into that role. The understanding at the end of that conversation was that Brent and I agreed that I could be worship leader that year, and then Brent could take over after I left.
I brought my guitar with me to camp that summer. Almost every night, I played and sang worship songs as my campers were supposed to be falling asleep. I co-led songs with Michael in front of campfire a few times. I took advantage of every opportunity I could get to practice my worship leading skills.
Then, in a communication that I had with my recently appointed youth pastor during late July, I was told that Brent was worship leader. I ignored the issue for weeks. Brent has zero singing ability and he's a sophomore--of course he's not worship leader. I kept thinking that there was going to be an application process, an open audition for those who wanted the position.
It was September 2010. I was a month into my senior year at Berean. I had been playing on the youth group worship team nearly every week. There was no audition.
I had been snubbed.
I later learned that over the previous year, Cory had apparently been "training" and "grooming" Brent--who was a freshman--to take over for him. He had never approached me about the issue. Conversely, I was unaware that I needed to ask him about it. Evidently, worship leadership was an insiders club that was invite-only.
And, evidently, Brent went back on our soft agreement that I could be worship leader that year.
So all that time I spent practicing was for naught? I thought. Why did I bring my guitar to camp?
I had been snubbed.
Berean
Jonathan's worship leading was mediocre. His team was the worst out of the four teams I had seen during my time at Berean. Each week during chapel, I identified something he did that either was a mistake or that I disliked. And each week, it did not get any less painful for me.
In October 2010, I was notified that there was a leadership-based scholarship application open. Only one student from Berean would be permitted to apply, so I completed a "pre-application." The College and Career Counselor would use these pre-applications to decide who would actually get to submit the application. Jonathan and I were the two pre-application finalists. The Counselor met with both of us together. She said that we were equally qualified; however, they gave the opportunity to apply to Jonathan because he had the official title of "worship leader."
I had been snubbed.
Church
I felt guilty. Had my parents wasted money getting me that guitar for Christmas?
I desperately held onto the wildest imagination of hope I could come up with: that if I talked to my youth pastor, he would replace Brent and put me in the position for which I had pined for so long. That hope was dashed to pieces in November when I heard the youth pastor describe his vision of having Brent as worship leader for the next three years of his high school career. He stated his intention of watching Brent grow and improve in the position. Why would you want a crappy worship band now, I thought, when I can already sing?
Berean
I spoke with a school friend of mine about my struggle to get over not being selected for worship leader. I asked her if she thought it would be worthwhile for me to talk to the student government advisor to get closure.
"Do you know why they chose Jonathan?"
"No."
"Student government said that during his audition, they felt the Holy Spirit in the room."
That changed the game for me. It broke my assumption that worship leader was what God wanted for me. Why did no one tell me this? I thought. That could have saved me six months of anguish.
Church
In December 2010, I told my youth pastor that I was quitting the worship team. I couldn't take it anymore. It was too painful.
Seriously, had my parents wasted money getting me that guitar?
* * *
I wish there were a better way to end this story. But any ending that neatly ties the story together and resolves everything is merely fictional.
"Well, hey...I could do it...if I just learned guitar..."
It was September 2009. I had recently begun my junior year at Berean Christian High School. I could play a G, a C, and a D on guitar. I knew how to make a bar chord (saying nothing about my ability to actually do so). And I was at least halfway decent at singing--hell, I was in the top choir at the school.
The idea of being worship leader was so exciting to me that I immediately pulled Grandpa Chet's old Martin guitar--the one that was apparently worth $X,000--out of the closet to start tuning and playing chords. My brother had left a piece of paper somewhere in our room which depicted the finger placement on the frets necessary to make common chords. It'll take a lot of work, I thought, but I have a year. I could do it.
Besides, I thought, even if it doesn't work out, I will have learned how to play guitar. And I could be worship leader for my church youth group.
Berean
The next few months, my mind was occupied with ideas and plans for how I would conduct chapels at school if I were worship leader. I closely observed what Melanie--the figurehead worship leader--and Travis--the actual worship leader--did, and I evaluated which aspects I wanted to keep or throw out. Every time my mind wandered to this topic, an overwhelming feeling of excitement welled up in me.
In November 2009, I scheduled an informal worship night at my church. It was intended to be a trial run for how my worship leading might go. I took great care in planning the set list and practicing for the event. I must have invited thirty or more people. Two showed up. Needless to say, it was disheartening.
But I kept on. I knew one failed event couldn't be what stopped me.
I needed an acoustic-electric guitar. The Martin didn't have a pickup, and there was no way my mom would have let me lug that guitar to and from school for chapel once a week anyway. I went to Guitar Center and tried various models out. I decided I liked best a black Ovation with a curved back. It wasn't too expensive ($300), but it also wasn't the cheapest, so I felt good about that. I added it to my Christmas list. Part of me felt guilty for asking for it since I was still far from an expert at guitar.
My parents bought me the Ovation for Christmas. I liked how it looked and how it felt; however, it had some pitch issues. Adjacent strings could be in pitch with each other at the fifth fret; but when the strings were open, each string got progressively more sharp. A source of the issue was the unusually high action. A Guitar Center employee helped me remove a half-millimeter-thick piece of plastic from under the saddle, which helped, but did not completely fix the issue.
Church
One Saturday in February of 2010, I showed up to my youth group's worship team practice to provide vocals. My goal was twofold: to break the stereotype of female backup singers in youth group bands, and to demonstrate (both to myself and to the group) that I could sing and had no problem doing so in front of the youth group.
Later that month, I accompanied a friend of mine who sang at a funeral at my church. This event not only gave me more practice, but I also viewed it as a marketing effort to my peers at my church that I could be my youth group's worship leader.
Berean
A couple weeks later, on a rainy afternoon after school, I spoke with Melanie and told her about my intention to audition for worship leader. I asked her some questions about her experience in the position, and she asked me about my interest. At the end of the conversation, she told me that a few people had come to her directly and told her that I would make a great worship leader. I was both baffled and encouraged. I had told very few people about my interest in the position; but it was also a good sign that my peers were endorsing me on their own.
It was Tuesday, March 30, 2010. I skipped the lunchtime musical practice to audition for worship leader in front of the student government. In addition to the usual speech, this year's audition also required candidates to lead the student government in a song. I gave a five-minute speech which I had loosely prepared for in which I explained where my desire came from and what I wanted to implement as worship leader. Then I led the student government in "Jesus Paid it All". I left the audition feeling fine. It was nothing spectacular, but as I thought, it was all in God's hands then.
On Friday, April 2, Good Friday, I was supposed to give "Noticias del Día" in Period 1 Spanish IV. I was nervous--not because I had to speak in Spanish in front of fifteen of my classmates, but because they were announcing the results of the student government elections on the intercom with the morning announcements.
"Worship leader is Jonathan Lim."
My heart did not stop pounding, even as the announcements finished and I walked up to the front of the room. My friend Curtis mouthed to me, "I'm sorry." I mouthed back, "It's fine."
At break, Dawn and I were chatting. Apparently she had missed the announcement. She was one of the ones that knew I wanted to go for worship leader. When she heard that Jonathan had won the position, she was indignant. "What!? You could do such a better job!" She was more upset about it than I was.
Melanie found me later in the day. She said it was neither my speech nor my song that didn't get me the position. She still wanted me to be involved with the worship team, so she asked me to audition for the worship team. I declined. I didn't perceive myself as good enough at guitar alone or at singing alone to provide only one of those talents. I could provide both, together, halfway decently, coupled with the leadership aspect, and that was all I was willing to give. I also knew that Mrs. Hazelrigg--the choir director--Melanie, and Travis--the favorite son of the music department--would be involved in deciding who comprised the worship team, and I did not want to be subject to their scrutiny.
The more I thought about it, the more upset I became. Jonathan had never demonstrated any evidence that he had any musical ability. Conversely, I had been in the school's top choir, the Abundant Life Singers, for two years. Historically, the worship leader had never not been a member of ALS, probably because second period (when ALS was scheduled) always followed chapel, so the worship leader could return all the musical equipment to the music room and not miss a more academic class.
But most of my unhappiness came from the fact that I had managed to convince myself that worship leader was what God wanted for me. I thought I had seen so many signs pointing to that path. I began to think that the student government went against what God wanted. Enmity toward the student government, toward Jonathan, and toward Melanie began to build within me.
Church
I took some solace in the thought that I could still lead worship at my youth group.
I had sung in the church's youth group band twice. The first time was when I showed up on my own, and the second time was the following week, when Cory, the worship leader asked me to return. I was never scheduled to return again. Each week, I received an email from Cory with the list of musicians for the following Sunday. And each week, my name was not on the list. It bothered me that I was not able to demonstrate my ability and interest in the worship team; but I also knew that the youth group didn't have much of a choice for the following year's worship leader selection. The current worship band consisted entirely of graduating seniors, with the exception of one terrible vocalist and one electric guitar player, Brent. So I continued refining my skills on guitar, even though I had been rejected for the position at school.
Berean
One evening in June 2010, I had a hard time coping with my Berean worship leader defeat. I wanted a reason. So I reached out to Melanie on Facebook. I asked her what it was that disqualified me, if it was neither my speech nor my song. Her response, saturated with agitation, was that it was nothing personal against me; that student government collectively decided that Jonathan would be a better fit; and that I should talk to the student government adviser, because it was not her place to say.
Less than a week later, she deleted me as a Facebook friend.
Church
One Sunday in early June 2010, Cory was out of town. This is my chance to step up, I thought. So I offered to lead the youth group worship that Sunday. Cory left behind no instructions, no guidance, no access to sheet music. So of course none of the electrical equipment worked that morning, and I was left scrambling to figure out what to do in front of my peers and my youth pastor. It felt as though I had been deliberately set up for failure.
Brent and I got together another Saturday that summer to practice for Sunday's worship. As we were leaving his house, his mom called out, suggesting that I lead worship for the youth group during the school year. She said it seemed logical, as I was graduating and Brent was only an incoming sophomore. I admitted that I had been hoping to step into that role. The understanding at the end of that conversation was that Brent and I agreed that I could be worship leader that year, and then Brent could take over after I left.
I brought my guitar with me to camp that summer. Almost every night, I played and sang worship songs as my campers were supposed to be falling asleep. I co-led songs with Michael in front of campfire a few times. I took advantage of every opportunity I could get to practice my worship leading skills.
Then, in a communication that I had with my recently appointed youth pastor during late July, I was told that Brent was worship leader. I ignored the issue for weeks. Brent has zero singing ability and he's a sophomore--of course he's not worship leader. I kept thinking that there was going to be an application process, an open audition for those who wanted the position.
It was September 2010. I was a month into my senior year at Berean. I had been playing on the youth group worship team nearly every week. There was no audition.
I had been snubbed.
I later learned that over the previous year, Cory had apparently been "training" and "grooming" Brent--who was a freshman--to take over for him. He had never approached me about the issue. Conversely, I was unaware that I needed to ask him about it. Evidently, worship leadership was an insiders club that was invite-only.
And, evidently, Brent went back on our soft agreement that I could be worship leader that year.
So all that time I spent practicing was for naught? I thought. Why did I bring my guitar to camp?
I had been snubbed.
Berean
Jonathan's worship leading was mediocre. His team was the worst out of the four teams I had seen during my time at Berean. Each week during chapel, I identified something he did that either was a mistake or that I disliked. And each week, it did not get any less painful for me.
In October 2010, I was notified that there was a leadership-based scholarship application open. Only one student from Berean would be permitted to apply, so I completed a "pre-application." The College and Career Counselor would use these pre-applications to decide who would actually get to submit the application. Jonathan and I were the two pre-application finalists. The Counselor met with both of us together. She said that we were equally qualified; however, they gave the opportunity to apply to Jonathan because he had the official title of "worship leader."
I had been snubbed.
Church
I felt guilty. Had my parents wasted money getting me that guitar for Christmas?
I desperately held onto the wildest imagination of hope I could come up with: that if I talked to my youth pastor, he would replace Brent and put me in the position for which I had pined for so long. That hope was dashed to pieces in November when I heard the youth pastor describe his vision of having Brent as worship leader for the next three years of his high school career. He stated his intention of watching Brent grow and improve in the position. Why would you want a crappy worship band now, I thought, when I can already sing?
Berean
I spoke with a school friend of mine about my struggle to get over not being selected for worship leader. I asked her if she thought it would be worthwhile for me to talk to the student government advisor to get closure.
"Do you know why they chose Jonathan?"
"No."
"Student government said that during his audition, they felt the Holy Spirit in the room."
That changed the game for me. It broke my assumption that worship leader was what God wanted for me. Why did no one tell me this? I thought. That could have saved me six months of anguish.
Church
In December 2010, I told my youth pastor that I was quitting the worship team. I couldn't take it anymore. It was too painful.
Seriously, had my parents wasted money getting me that guitar?
* * *
I wish there were a better way to end this story. But any ending that neatly ties the story together and resolves everything is merely fictional.
Sunday, September 10, 2017
John Piper: line by line deconstruction
John Piper is an esteemed Christian theologian, especially by those in evangelical circles. He founded the organization and website Desiring God. He is a contributor to The Gospel Coalition. I think North Creek Church, a local church which informed much of the theology I was taught in high school, treats him more or less in the same way that the Roman Catholic Church treats the Virgin Mary.
He is also literally the worst.
He recently gave this abhorrent talk. Fortunately, it had been transcribed so I didn't have to suffer through hearing someone actually say the words aloud. There were so many points of contention that I thought it merited a post for me to organize my thoughts on the content.
Here we go.
"If you know God, and his greatness, and his wrath, holiness, justice, wrath, grace, you will tremble in his presence."
Okay, so I actually did press play to hear this to see if "wrath" was really said twice, or if that was just a typo. Nope, he really said it twice, the second instance a mere three words apart from the first within the same list. Maybe he just accidentally re-read a line of his sermon notes, but that's being gracious. More likely, it wasn't a mistake, and it communicates something about his underlying view of who God is.
"Mainly, I’m talking to dads, and the moms who have to play Dad the best you can: get other men around you, who can help you do it."
Translation: Ladies, you don't know how to raise your own children. Find a man--literally, my only criterion is that this person must be a man--to teach kids to live in perpetual fear. It'll be great, I swear.
"Dads, help your children tremble with joy in the presence of God. I have one means to suggest for how to do it. Fathers, be the kind of father that your children delight to fear."
"Tremble with joy." That statement by itself maybe could be stretched to mean "help your children be so in love with God that they tremble with joy and excitement to be in his presence." Except that's not what he's talking about. He's talking about trembling with fear. Which, believe it or not Mr. Piper, cannot coincide with joy. Just as "delight to fear" is also an oxymoron. There is a verse in your holy book that says "There is no fear in love. But perfect love casts out fear". What do you do with that one?
"I am God to my children until they know better."
Um, what? This kind of thinking got all kinds of people in the Bible into trouble. Like the King of Babylon (or Satan/Lucifer, depending on how you interpret that passage), the King of Tyre, the Pharaoh of Egypt, James and John, and this dude here.
"If they only fear you and there’s no delight in it, it’s wrong and dysfunctional. If they only delight in you and there’s no fear in it, it’s wrong and dysfunctional. In both cases, you have made it very difficult for them to embrace the true God."
As if parenthood isn't already hard enough by itself, you now have to ride this ridiculously fine line. Somehow, this is all I can think of:
"That’s a high calling, which means, there must be wrath and there must be compassion. The children must learn it."
Hey dads, you've just gotten a free pass to have your frustration with your kids be totally unchecked. Feel free to act on your anger and frustration; in fact, you are compelled to act upon them. That's definitely not going to mess up your kids' mental health at all. (Special note: this isn't the first article that I've read by Mr. Piper in which he legitimizes and encourages cultivating anxiety in your children.)
"Why? Because that’s the way God is. Hebrews 12:6: 'The Lord disciplines the one he loves, and chastises every son whom he receives.'"
Manipulating this verse in this way is quite triggering for me. It's the exact justification my Christian fraternity gave me for its abusive actions. The counterargument is simple: you're substituting "earthly fathers" (or "the fraternity") for "The Lord". That's kind of blasphemous. Although, apparently the implication of substituting yourself for "The Lord" isn't clear to Mr. Piper. See three points ago.
"That’s why little babies have fat bottoms."
LOLOLOLOLOLOLOL WHAT THE FUCK
No, but seriously, what the fuck!?
So Mr. Piper is an expert in the evolutionary processes that have made the human buttocks develop in the fashion that it does. And evidently, corporal punishment applied to the buttocks is the evolutionary function of babies' extra body fat. Assuming this is true, it begs the question: why not apply corporal punishment to a baby's lower legs? Or to their cheeks? Or to the backs of their necks? You know, other places where babies have large stores of fat.
Implicit in this statement is that the buttocks is where and how you should discipline your child.
"You also read in the Bible Ephesians 6:4: 'Fathers, do not provoke your children to anger, but bring them up in the discipline and instruction of the Lord.' Do you hear that phrase? 'Bring them up in the discipline of the Lord.' This is not yours, this is God's. You are playing God here. 'You are representing me there,' God says. 'So, bring them up in my discipline, in such a way that they don’t get angry, but delight in your discipline.'"
There's a disproportionate focus on the "discipline" aspect of the verse, rather than the "do not provoke your children to anger".
"Fathers are to represent God so that children will come to believe in hell and heaven and Christ through Dad."
Just going to point out that "hell" was listed before "heaven". And that the danger of hellfire was listed before the mention of Christ/grace.
"So, I think we want our children to delight to fear us. In a store--where of course if you spank them, you’ll get arrested--our eyes should be able to do it. Your eyes are like lightning ready to strike."
Make sure that your kids develop a negative Pavlovian response to your mere gaze. That's really healthy for a parent-child relationship.
And, of course, nevermind that underlying that statement is the idea that it is a shame that we can't spank our kids in public.
"I want them to say, 'Daddy, disobeying you is a fearful thing, and I am thankful that it is. But oh, what a sanctuary you are to me. How I love to live in the light of your strength and joy.'"
Twisted, twisted, twisted. Also, who the fuck talks like that?
I originally also had a retort to the final sentence of the article, but I can't remember what it is now. Oh well. I think I've covered all of my instinctive reactions that I had when I first read the article.
John Piper is insane, I know that. So why am I picking a fight with him? Because people in the church revere his evil statements, and religion grants him a blanket of protection for his destructive ideology. Until madmen are universally recognized as such, it is essential to explain to their adherents why they are madmen.
He is also literally the worst.
He recently gave this abhorrent talk. Fortunately, it had been transcribed so I didn't have to suffer through hearing someone actually say the words aloud. There were so many points of contention that I thought it merited a post for me to organize my thoughts on the content.
Here we go.
"If you know God, and his greatness, and his wrath, holiness, justice, wrath, grace, you will tremble in his presence."
Okay, so I actually did press play to hear this to see if "wrath" was really said twice, or if that was just a typo. Nope, he really said it twice, the second instance a mere three words apart from the first within the same list. Maybe he just accidentally re-read a line of his sermon notes, but that's being gracious. More likely, it wasn't a mistake, and it communicates something about his underlying view of who God is.
"Mainly, I’m talking to dads, and the moms who have to play Dad the best you can: get other men around you, who can help you do it."
Translation: Ladies, you don't know how to raise your own children. Find a man--literally, my only criterion is that this person must be a man--to teach kids to live in perpetual fear. It'll be great, I swear.
"Dads, help your children tremble with joy in the presence of God. I have one means to suggest for how to do it. Fathers, be the kind of father that your children delight to fear."
"Tremble with joy." That statement by itself maybe could be stretched to mean "help your children be so in love with God that they tremble with joy and excitement to be in his presence." Except that's not what he's talking about. He's talking about trembling with fear. Which, believe it or not Mr. Piper, cannot coincide with joy. Just as "delight to fear" is also an oxymoron. There is a verse in your holy book that says "There is no fear in love. But perfect love casts out fear". What do you do with that one?
"I am God to my children until they know better."
Um, what? This kind of thinking got all kinds of people in the Bible into trouble. Like the King of Babylon (or Satan/Lucifer, depending on how you interpret that passage), the King of Tyre, the Pharaoh of Egypt, James and John, and this dude here.
"If they only fear you and there’s no delight in it, it’s wrong and dysfunctional. If they only delight in you and there’s no fear in it, it’s wrong and dysfunctional. In both cases, you have made it very difficult for them to embrace the true God."
As if parenthood isn't already hard enough by itself, you now have to ride this ridiculously fine line. Somehow, this is all I can think of:
"That’s a high calling, which means, there must be wrath and there must be compassion. The children must learn it."
Hey dads, you've just gotten a free pass to have your frustration with your kids be totally unchecked. Feel free to act on your anger and frustration; in fact, you are compelled to act upon them. That's definitely not going to mess up your kids' mental health at all. (Special note: this isn't the first article that I've read by Mr. Piper in which he legitimizes and encourages cultivating anxiety in your children.)
"Why? Because that’s the way God is. Hebrews 12:6: 'The Lord disciplines the one he loves, and chastises every son whom he receives.'"
Manipulating this verse in this way is quite triggering for me. It's the exact justification my Christian fraternity gave me for its abusive actions. The counterargument is simple: you're substituting "earthly fathers" (or "the fraternity") for "The Lord". That's kind of blasphemous. Although, apparently the implication of substituting yourself for "The Lord" isn't clear to Mr. Piper. See three points ago.
"That’s why little babies have fat bottoms."
LOLOLOLOLOLOLOL WHAT THE FUCK
No, but seriously, what the fuck!?
So Mr. Piper is an expert in the evolutionary processes that have made the human buttocks develop in the fashion that it does. And evidently, corporal punishment applied to the buttocks is the evolutionary function of babies' extra body fat. Assuming this is true, it begs the question: why not apply corporal punishment to a baby's lower legs? Or to their cheeks? Or to the backs of their necks? You know, other places where babies have large stores of fat.
Implicit in this statement is that the buttocks is where and how you should discipline your child.
"You also read in the Bible Ephesians 6:4: 'Fathers, do not provoke your children to anger, but bring them up in the discipline and instruction of the Lord.' Do you hear that phrase? 'Bring them up in the discipline of the Lord.' This is not yours, this is God's. You are playing God here. 'You are representing me there,' God says. 'So, bring them up in my discipline, in such a way that they don’t get angry, but delight in your discipline.'"
There's a disproportionate focus on the "discipline" aspect of the verse, rather than the "do not provoke your children to anger".
"Fathers are to represent God so that children will come to believe in hell and heaven and Christ through Dad."
Just going to point out that "hell" was listed before "heaven". And that the danger of hellfire was listed before the mention of Christ/grace.
"So, I think we want our children to delight to fear us. In a store--where of course if you spank them, you’ll get arrested--our eyes should be able to do it. Your eyes are like lightning ready to strike."
Make sure that your kids develop a negative Pavlovian response to your mere gaze. That's really healthy for a parent-child relationship.
And, of course, nevermind that underlying that statement is the idea that it is a shame that we can't spank our kids in public.
"I want them to say, 'Daddy, disobeying you is a fearful thing, and I am thankful that it is. But oh, what a sanctuary you are to me. How I love to live in the light of your strength and joy.'"
Twisted, twisted, twisted. Also, who the fuck talks like that?
I originally also had a retort to the final sentence of the article, but I can't remember what it is now. Oh well. I think I've covered all of my instinctive reactions that I had when I first read the article.
John Piper is insane, I know that. So why am I picking a fight with him? Because people in the church revere his evil statements, and religion grants him a blanket of protection for his destructive ideology. Until madmen are universally recognized as such, it is essential to explain to their adherents why they are madmen.
Wednesday, September 6, 2017
White gays are the worst.
Right, so fairly recently, an incident at Toledo Pride occurred which made its rounds on the internet a couple weeks ago.
Ok, so maybe it was only a part of the internet that I see, but whatever.
The story goes like this: a black guy starts talking to a white guy on Grindr who is in an open relationship. At some point, the white guy's boyfriend takes his boyfriend's phone and tells the black guy...
You know what, I'll just let the barrage of Grindr messages speak for itself.
"Even if we were looking for a threesome, we would never hookup with someone like you… we don’t like dark skin, or dark burn culture. We are classy like America. We don’t twerk and do stupid shit that fat black people do. We are civilised people. Please don’t bring your burn culture into ours its already good thanks. Blacks are too dark and have ruined beauty for us queers. Take ur fried chicken and fatty foods elsewhere away from us. You nigers…you can’t even get a nice job. You all are so poor. Can’t even afford a new car. Black people have black skin and is nasty and discussing and unappealing. Stop talking to us whites. We don’t want dark or burnt skinned friends or sex partners… Blacks are unattractive and ugly to even look at… Like you don’t deserve to have the attention of a white man…. I don’t want my clean big dick in some dark boy pussy or black skinned mouth. I would feel ashamed and disgusted."
Source: PinkNews
So, the gay community has a problem with racism. What else is new?
I figured common sense would lead anyone to condemn this racist treatment. But no. Instead, this guy decides to voice his opinion on the matter.
Here's my summary of the article: Well, black guys are racist too!
Okay, maybe, but that doesn't mean that the above Grindr message was remotely okay. Nowhere was there any condemnation of the message. Instead, the writer seems to blame black men who are only into white guys for the message.
Yes, rigid sexual preferences that neither allow for exceptions nor are open to change ("I only date white guys" or "I only date uncut") should indeed be denounced. But they should be denounced on all sides, including the white asshole who authored the message above. Why is it that black men who pursue only white guys are critiqued, whereas white men who pursue only white guys are not, even though both have been socialized to automatically view white as desirable?
All I wanted was for the author of the response article to express disapproval for the author of the Grindr messages. Instead, the burden of fixing internalized racism falls on black men, rather than on those who have substantially more power and representation in the gay community.
And, I know I'm treading dangerous waters, as the guy who wrote that article is black, but you don't get to call black men out for racially biased sexual preferences and also call them "coons" in the same breath. Just, no.
Ok, so maybe it was only a part of the internet that I see, but whatever.
The story goes like this: a black guy starts talking to a white guy on Grindr who is in an open relationship. At some point, the white guy's boyfriend takes his boyfriend's phone and tells the black guy...
You know what, I'll just let the barrage of Grindr messages speak for itself.
"Even if we were looking for a threesome, we would never hookup with someone like you… we don’t like dark skin, or dark burn culture. We are classy like America. We don’t twerk and do stupid shit that fat black people do. We are civilised people. Please don’t bring your burn culture into ours its already good thanks. Blacks are too dark and have ruined beauty for us queers. Take ur fried chicken and fatty foods elsewhere away from us. You nigers…you can’t even get a nice job. You all are so poor. Can’t even afford a new car. Black people have black skin and is nasty and discussing and unappealing. Stop talking to us whites. We don’t want dark or burnt skinned friends or sex partners… Blacks are unattractive and ugly to even look at… Like you don’t deserve to have the attention of a white man…. I don’t want my clean big dick in some dark boy pussy or black skinned mouth. I would feel ashamed and disgusted."
Source: PinkNews
So, the gay community has a problem with racism. What else is new?
I figured common sense would lead anyone to condemn this racist treatment. But no. Instead, this guy decides to voice his opinion on the matter.
Here's my summary of the article: Well, black guys are racist too!
Okay, maybe, but that doesn't mean that the above Grindr message was remotely okay. Nowhere was there any condemnation of the message. Instead, the writer seems to blame black men who are only into white guys for the message.
Yes, rigid sexual preferences that neither allow for exceptions nor are open to change ("I only date white guys" or "I only date uncut") should indeed be denounced. But they should be denounced on all sides, including the white asshole who authored the message above. Why is it that black men who pursue only white guys are critiqued, whereas white men who pursue only white guys are not, even though both have been socialized to automatically view white as desirable?
All I wanted was for the author of the response article to express disapproval for the author of the Grindr messages. Instead, the burden of fixing internalized racism falls on black men, rather than on those who have substantially more power and representation in the gay community.
And, I know I'm treading dangerous waters, as the guy who wrote that article is black, but you don't get to call black men out for racially biased sexual preferences and also call them "coons" in the same breath. Just, no.
Sunday, August 13, 2017
A Smorgasbord of Thoughts on God and Religion.
It's time to call a spade a spade. The times in my life when I felt that my relationship with God was at it strongest completely overlapped with the periods when I had the most destructive views of myself. The converse is also true; I felt most distant from God when I actually had a decent sense of self-worth.
Here are some examples.
1) I felt close to God in 2012 during and following my fraternity pledgeship. I interpreted the trauma inflicted upon me as God testing me and growing my faith, rather than sticking up for myself and asserting to the fraternity, "This cannot happen."
2) I felt distant from God in Fall 2013 after months of counseling. In these counseling sessions--which were with a Christian therapist, by the way--I learned to accept my thoughts and feelings as legitimate and worthy of others' considerations.
3) Since Fall 2013, the health of my self-perception has increased. And I haven't felt "close" to God since then.
I suppose when "God" is defined in the way that he was for me throughout most of my life, you inevitably must devalue yourself in order to have a relationship in good standing with him.
* * *
Evangelical Christianity is powerful in the sense that it exerts dominion over every aspect of your life. Under such a belief system, you're not allowed to quarantine your faith to a religious compartment of your life. Literally everything that you think, say, and do is subject to an audit. It's exhausting.
I marveled at the many people I knew in college who professed Catholicism. Their faith dictated little more in their lives than what they did on Sunday. Beyond that, they had a freedom far greater than anything that I was used to. And that in itself made me skeptical that their faith was real.
* * *
"You know, I rather like this God fellow. Very theatrical, you know. Pestilence here, a plague there. Omnipotence! Gotta get me some of that." -- Stewie Griffin
* * *
What is the rate of conversion to Christianity (or any faith, really) after the age of, say, 20? My guess is that it's tiny. Certainly a far smaller rate than the same rate sampled before the age of 10.
The point is this: How much of an indicator of one's belief system is the belief system of his or her parents?
What if my parents had been Buddhists? Or Muslims? Or atheists? It seems very unlikely to me that I would have found and willingly chosen to convert to Christianity under these conditions.
* * *
If we really believe the words of the Bible, then God should be largely unknowable. Frankly, that's kind of what chapters 38 through 41 of the book of Job seem to imply.
That morality seems to be present in all people was somewhat of a startling realization for me when I got to college. Religion does not, in fact, hold a monopoly on morality; irreligious people are generally no more immoral than anyone else. For a while, I took that to mean that some form of God is present in everyone. That in turn invalidated a core pillar of my belief system growing up--that people are basically evil.
So then at what point does the use of the word "God" become arbitrary? If a supernatural being has the above characteristics (unknowable, yet manifested in everyone), what makes it "God" versus some generic cosmic energy, or even natural selection?
* * *
"Oh, I love God! He's so deliciously evil!" -- Stewie Griffin
* * *
Defaulting to a God hypothesis to explain things in the universe that we don't understand should not satisfy anyone. I don't say this because I think lowly of such a hypothesis; I say this because such a hypothesis discourages seeking to understand the world further. There is no need to learn, to study, or to think if every problem can be satisfied by, "God made it this way." What a waste of a human brain that is.
Here are some examples.
1) I felt close to God in 2012 during and following my fraternity pledgeship. I interpreted the trauma inflicted upon me as God testing me and growing my faith, rather than sticking up for myself and asserting to the fraternity, "This cannot happen."
2) I felt distant from God in Fall 2013 after months of counseling. In these counseling sessions--which were with a Christian therapist, by the way--I learned to accept my thoughts and feelings as legitimate and worthy of others' considerations.
3) Since Fall 2013, the health of my self-perception has increased. And I haven't felt "close" to God since then.
I suppose when "God" is defined in the way that he was for me throughout most of my life, you inevitably must devalue yourself in order to have a relationship in good standing with him.
* * *
Evangelical Christianity is powerful in the sense that it exerts dominion over every aspect of your life. Under such a belief system, you're not allowed to quarantine your faith to a religious compartment of your life. Literally everything that you think, say, and do is subject to an audit. It's exhausting.
I marveled at the many people I knew in college who professed Catholicism. Their faith dictated little more in their lives than what they did on Sunday. Beyond that, they had a freedom far greater than anything that I was used to. And that in itself made me skeptical that their faith was real.
* * *
"You know, I rather like this God fellow. Very theatrical, you know. Pestilence here, a plague there. Omnipotence! Gotta get me some of that." -- Stewie Griffin
* * *
What is the rate of conversion to Christianity (or any faith, really) after the age of, say, 20? My guess is that it's tiny. Certainly a far smaller rate than the same rate sampled before the age of 10.
The point is this: How much of an indicator of one's belief system is the belief system of his or her parents?
What if my parents had been Buddhists? Or Muslims? Or atheists? It seems very unlikely to me that I would have found and willingly chosen to convert to Christianity under these conditions.
* * *
If we really believe the words of the Bible, then God should be largely unknowable. Frankly, that's kind of what chapters 38 through 41 of the book of Job seem to imply.
That morality seems to be present in all people was somewhat of a startling realization for me when I got to college. Religion does not, in fact, hold a monopoly on morality; irreligious people are generally no more immoral than anyone else. For a while, I took that to mean that some form of God is present in everyone. That in turn invalidated a core pillar of my belief system growing up--that people are basically evil.
So then at what point does the use of the word "God" become arbitrary? If a supernatural being has the above characteristics (unknowable, yet manifested in everyone), what makes it "God" versus some generic cosmic energy, or even natural selection?
* * *
"Oh, I love God! He's so deliciously evil!" -- Stewie Griffin
* * *
Defaulting to a God hypothesis to explain things in the universe that we don't understand should not satisfy anyone. I don't say this because I think lowly of such a hypothesis; I say this because such a hypothesis discourages seeking to understand the world further. There is no need to learn, to study, or to think if every problem can be satisfied by, "God made it this way." What a waste of a human brain that is.
Friday, July 7, 2017
Prisoner Problem: A Riddle (Post 2 of 2) SPOILERS
IF YOU READ THE PREVIOUS POST AND WANT TO FIND THE SOLUTION ON YOUR OWN, DO NOT READ THIS POST.
Actually, I have no idea if anyone actually reads this blog anyway, so I might just be shouting at nothing.
In any case, here.
PART 1: The Solution
The prisoners will choose one of them to be the "spokesperson" for the group. Let's call this prisoner the principal. The prisoners all decide that the principal is the only one who can offer the answer "Yes" to the warden; all other prisoners will always say "No" until the challenge is complete.
The switches both start in the down/off position. Let's call the left switch Switch 1 and the right switch Switch 2. Switch 1 will be the indicator switch, and Switch 2 will be the "dummy" switch.
When a prisoner is selected, there are two possible cases that matter: whether or not the prisoner is the designated principal.
CASE 1: The prisoner selected is not the principal. The prisoner will ask himself, "Have I turned Switch 1 'on' before in this challenge?" If the answer is yes, then he will toggle Switch 2. If the answer is no, he will then examine Switch 1. If Switch 1 is already on, then he will toggle Switch 2. If Switch 1 is not on, then he will toggle Switch 1 to the on position. If this prisoner turned Switch 1 on, then during any subsequent visits to the room, this prisoner will only toggle Switch 2. Every prisoner who is not the principal must always respond "No" to the warden's question.
CASE 2: The prisoner selected is the principal. The principal is also tasked with the job of keeping a cumulative count. When he enters the room, he will examine Switch 1. If Switch 1 is on, he will add 1 to the cumulative count and will toggle Switch 1 to the off position. If Switch 1 is not on, he will add 0 to the cumulative count and will toggle Switch 2. Naturally, if it is the principal's first time to the room, he will also add an additional 1 to the cumulative count for himself. When the principal's cumulative count reaches n, the number of prisoners, then he will answer "Yes" to the warden's question.
This strategy takes a stochastic number of days to complete, which frustrated me at first. Essentially, Switch 1 being on means that a new prisoner has had the opportunity to turn Switch 1 on. (Switch 2 means nothing.) So the one person who is designated to check Switch 1 knows with certainty that n unique prisoners have been in the cell when he arrives to the room to an on Switch 1 for the (n - 1)th time. (The principal is obviously the last case.)
PART 2: Distribution Analysis
Two questions immediately popped into my head: How can the distribution of days that it will take to complete this challenge be characterized (specifically, what are the mean and variance of the number of days it will take)? How sensitive is this distribution to n, the number of prisoners?
Before answering these questions, we need a definition of a common discrete probability distribution.
The geometric distribution is a discrete distribution that counts the number of independent Bernoulli trials (essentially binary "success"/"fail" trials) that must be completed until a "success" is reached. A common example of a geometric distribution is, "How many flips of a coin does it take to get a 'heads'?" In this example, the trials are coin flips, and a "success" is getting a heads. Because the trials are assumed to be independent, each trial has the same success probability, denoted as p. In the coin-flip example, p = 1/2 if the coin is fair.
If a random variable X has a geometric distribution, we denote it as X ~ geom(p).
Here are two properties about the geometric distribution that we care about for this problem:
The mean of X, E(X), is 1/p. This value represents the expected number of trials needed to be conducted until a "success" is achieved. In the coin-flip example, we expect that on average, it will take two flips to get a "heads".
The variance of X, var(X), is (1-p)/p2
We will use these properties in determining the mean and the variance of the solution to this riddle.
PART 2a: Finding the Mean
Let's start with finding E(Tn), the expected number of days it will take until a solution is reached, as a function of n, the number of prisoners. Let's start with the case of n = 2 just to simplify the problem.
n = 2
For simplicity's sake, let's suppose the two prisoners have selected Prisoner 1 as the principal. (It doesn't actually matter which prisoner is selected to be the principal.) On day 1, the warden selects one of the two at random. At this point, the amount of time until a new prisoner can toggle Switch 1 follows a geometric distribution with probability 1 (E(X) = 1, Var(X) = 0). This first selection is critical; there are two possible sample paths that can follow.
CASE 1: The first prisoner selected is the principal. When the principal leaves, he sees Switch 1 as down, so he knows the other prisoner has not been in yet. He can add 1 to his cumulative count. Once he returns, the amount of time until Prisoner 2 enters the room follows a geometric distribution with probability 1/2. (E(X) = 2). Then, once Prisoner 2 has been in, the principal must go back in again to receive the message through Switch 1 that Prisoner 2 has been in. The amount of time that this takes again follows a geometric distribution with probability 1/2. At that point, the principal can answer the warden's question correctly with a "Yes", and all prisoners can go free. On average, this will take 1 + 2 + 2 = 5 days.
CASE 2: The first prisoner selected is not the principal. Prisoner 2 flips Switch 1 up. The next "success" is when the principal enters the room. The amount of time that this takes again follows a geometric distribution with probability 1/2. When the principal enters, he will see Switch 1 up, and he immediately knows that all prisoners have been in since he has also been in. On average, this will take 1 + 2 = 3 days.
Now we weight these two lengths of time by the probability of each case occurring. Since selection is equally likely and there are two prisoners, each case can occur with probability 1/2. Therefore, the overall mean length of time that it will take for n = 2 prisoners to get out is 0.5*5 + 0.5*3 = 4 days. So E(T2) = 4.
Now let us consider the case of n = 3 with the same logic.
n = 3
On day 1, the amount of time until a new prisoner can toggle Switch 1 is geometric with probability 1. However, we must again condition on which prisoner was chosen first.
CASE 1: The first prisoner selected is the principal. He will add 1 to his count. After he leaves, the amount of time until a new prisoner enters the room and can toggle Switch 1 is geometric with probability 2/3. After that, the principal must be selected again; the time this takes is geometric with probability 1/3. At this point, there is one prisoner left who must toggle Switch 1 for the first time, which will occur after an amount of time defined again by a geometric distribution with probability 1/3. The principal must once again be selected, adding on another geometric with probability 1/3. Adding up the mean of all these distribution gives an expected time of 1 + 3/2 + 3 + 3 + 3 = 11.5 days.
CASE 2: The first prisoner selected is not the principal. After this first prisoner, the next time Switch 1 will be touched is when the principal enters. This happens after geom(1/3) time. At that point, the principal can add 2 to his cumulative count. After he leaves, it is another geom(1/3) until a new prisoner can turn on Switch 1. Then the principal must be called again, which occurs after geom(1/3) time. This gives an expected time of 1 + 3 + 3 + 3 = 10 days.
Now we weight these two cases. Case 1 occurs with probability 1/3, and Case 2 occurs with probability 2/3. Therefore, the overall mean length of time that it will take n = 3 prisoners to get out is (1/3)*(23/2) + (2/3)*(10) = 10.5 days.
n = 4 [Abridged, but following the same logic.]
CASE 1: (Principal first, occurs with probability 1/4)
geom(1) + geom(3/4) + geom(1/4) + geom(2/4) + geom(1/4) + geom(1/4) + geom(1/4) yields a mean of 61/3 days.
CASE 2: (Principal not first, occurs with probability 3/4)
geom(1) + geom(1/4) + geom(2/4) + geom(1/4) + geom(1/4) + geom(1/4) yields a mean of 57/3 days.
The overall mean length of time that it will take n = 4 prisoners to get out is 58/3 days, or 19.333 days.
And so we can see a pattern forming. For a general n, we can condition on whether or not the first prisoner selected is the principal.
CASE 1: (Principal first, occurs with probability 1/n)
geom(1) + geom((n-1)/n) + geom(1/n) + geom((n-2)/n) + geom(1/n) + ... + geom(2/n) + geom(1/n) + geom(1/n) + geom(1/n). The mean of the sum of these random variables can be written as follows:
CASE 2: (Principal not first, occurs with probability (n-1)/n)
geom(1) + geom(1/n) + geom((n-2)/n) + geom(1/n) + geom((n-3)/n) + geom(1/n) + ... + geom(2/n) + geom(1/n) + geom(1/n) + geom(1/n). The mean of the sum of these random variables can be written as follows:
The mean for Case 2 is written in this way to simplify the sum. In Case 2, the principal counts 2 when he first arrives as opposed to just 1 in Case 1. So Case 2 does not have the geom((n-1)/n) term, but otherwise, the means are identical. So we can simply subtract off n/(n-1).
When we weight the mean from Case 1 by 1/n and the mean from Case 2 by (n-1)/n, we get the following beautifully simple result:
We see that the mean is quadratic and increasing in n, the number of prisoners. Plotted below is the mean number of days that will pass until the prisoners can all be released, as a function of n (from 1 to 15):
PART 2b: Finding the Variance
Of course, in practice, the mean, or expected value, rarely actually occurs. In most cases, such as when n = 3, the expected value is actually not a possible value that the distribution can take, since the length of time will always be an integer number of days.
Therefore, we may be interested in knowing how "spread out" the distribution is for the time until release. Naturally, we expect the distribution to be more spread out, or more variable, as n increases. But how can we precisely calculate the variance of this strategy?
We will make extensive use of the fact that the time until escape is a sum of geometric random variables, for which we know the variance. We will also make use of the fact that each prisoner is randomly selected. What this means is that all individual trials are independent of each other (hence we can use a geometric distribution); but furthermore, we can assume that all geometric distributions in the sum are independent of each other. This is a critical assumption, as the equation
Var(X + Y) = Var(X) + Var(Y)
in general does NOT hold unless the random variables X and Y are independent of each other. Since all of our geometric random variables are independent, we can in fact use this equation to our advantage.
Variance is defined as follows:
Var(X) = E[(X - E(X))2]
Verbally, this means that the variance of X is defined as the expected squared deviation from the mean of the X. This value is squared to ensure that the deviation is represented as a positive value; it unfortunately places disproportionate weight on large deviations.
As an aside: the phrase "standard deviation" is used in common culture frequently. The standard deviation is simply the square root of the variance. By Chebyshev's Inequality, for every distribution, at least 75% of a distribution falls within 2 standard deviations of the mean, and at least 88.9% of a distribution falls within 3 standard deviations of the mean. As a general rule of thumb, if a distribution is roughly bell-shaped, then at least 95% (or more) of the distribution will be within 3 standard deviations of the mean. That is why the standard deviation is significant.
Unfortunately, in this problem, calculating the variance is not straightforward, as it again depends on which type of prisoner is selected first. We unfortunately cannot just condition on what happens first and then weight the variance of each case by the probability of each first selection; instead, we must use the formula for conditional variance and condition on the first selection.
Var(X) = E(Var(X | Y)) + Var(E(X | Y))
It is important to note that E(X | Y) is a random variable, as is Var(X | Y). Otherwise, taking the variance and mean of these values, respectively, would not make sense.
We define the random variable Y as follows:
We could have defined it the other way around, but it does not matter. What matters is what value X, or in our case, Tn, takes on when Y is 1 or 0. Fortunately, from the exercise in calculating the mean, we know what possible values this random variable can take. We will use g() to denote a geometric random variable.
We already know what E(Tn | Y) is. This random variable's possible values are the means that were calculated above. Case 1 represents E(Tn | Y = 1) and Case 2 represents E(Tn | Y = 0). Consequently, we also know E(E(Tn | Y)), which by the definition of conditional expectation, is just E(Tn). We can use these values to calculate the second part of the conditional variance formula, Var(E(Tn | Y)). This will be done by using the basic definition of variance, as defined above (expected squared deviation from the mean).
After a whole bunch of terms cancelling out, this simply reduces to:
Intuitively, this should make sense. As n grows large, the difference in possible values for E(Tn | Y) asymptotically and decreasingly approaches 1. Additionally, the probability of one of the two values occurring approaches 1. Both of these factors explain why the variance of E(Tn | Y) is decreasing in n.
Now we must calculate E(Var(Tn | Y)). This is where we can weight the variances that arise based on which prisoner is selected first. We must take the variance of the two possible outcomes for Tn. This is fairly simple by the formula for the variance of a geometric random variable, although it is a mess to write out.
I hope you're good at rearranging your fractions, because nicely enough, it simplifies to this:
Recall that we must add E(Var(Tn | Y)) and Var(E(Tn | Y)) to find Var(Tn). When we do that, the 1/(n-1) terms cancel out, and we are left with the following formula for the variance:
This result is also strikingly simple. The variance is cubic in n, implying that increasing n has a greater impact on the variance than on the mean, which intuitively makes sense. Plotted below is the variance of the time to be released, as a function of n (from 1 to 15):
Now, for you naysayers out there, I also built a simulation to model this game strategy with different numbers of n to verify if these results were correct. They are.
Anyways. I had a good nerding out about this. Hope you enjoyed this problem.
Actually, I have no idea if anyone actually reads this blog anyway, so I might just be shouting at nothing.
In any case, here.
PART 1: The Solution
The prisoners will choose one of them to be the "spokesperson" for the group. Let's call this prisoner the principal. The prisoners all decide that the principal is the only one who can offer the answer "Yes" to the warden; all other prisoners will always say "No" until the challenge is complete.
The switches both start in the down/off position. Let's call the left switch Switch 1 and the right switch Switch 2. Switch 1 will be the indicator switch, and Switch 2 will be the "dummy" switch.
When a prisoner is selected, there are two possible cases that matter: whether or not the prisoner is the designated principal.
CASE 1: The prisoner selected is not the principal. The prisoner will ask himself, "Have I turned Switch 1 'on' before in this challenge?" If the answer is yes, then he will toggle Switch 2. If the answer is no, he will then examine Switch 1. If Switch 1 is already on, then he will toggle Switch 2. If Switch 1 is not on, then he will toggle Switch 1 to the on position. If this prisoner turned Switch 1 on, then during any subsequent visits to the room, this prisoner will only toggle Switch 2. Every prisoner who is not the principal must always respond "No" to the warden's question.
CASE 2: The prisoner selected is the principal. The principal is also tasked with the job of keeping a cumulative count. When he enters the room, he will examine Switch 1. If Switch 1 is on, he will add 1 to the cumulative count and will toggle Switch 1 to the off position. If Switch 1 is not on, he will add 0 to the cumulative count and will toggle Switch 2. Naturally, if it is the principal's first time to the room, he will also add an additional 1 to the cumulative count for himself. When the principal's cumulative count reaches n, the number of prisoners, then he will answer "Yes" to the warden's question.
This strategy takes a stochastic number of days to complete, which frustrated me at first. Essentially, Switch 1 being on means that a new prisoner has had the opportunity to turn Switch 1 on. (Switch 2 means nothing.) So the one person who is designated to check Switch 1 knows with certainty that n unique prisoners have been in the cell when he arrives to the room to an on Switch 1 for the (n - 1)th time. (The principal is obviously the last case.)
PART 2: Distribution Analysis
Two questions immediately popped into my head: How can the distribution of days that it will take to complete this challenge be characterized (specifically, what are the mean and variance of the number of days it will take)? How sensitive is this distribution to n, the number of prisoners?
Before answering these questions, we need a definition of a common discrete probability distribution.
The geometric distribution is a discrete distribution that counts the number of independent Bernoulli trials (essentially binary "success"/"fail" trials) that must be completed until a "success" is reached. A common example of a geometric distribution is, "How many flips of a coin does it take to get a 'heads'?" In this example, the trials are coin flips, and a "success" is getting a heads. Because the trials are assumed to be independent, each trial has the same success probability, denoted as p. In the coin-flip example, p = 1/2 if the coin is fair.
If a random variable X has a geometric distribution, we denote it as X ~ geom(p).
Here are two properties about the geometric distribution that we care about for this problem:
The mean of X, E(X), is 1/p. This value represents the expected number of trials needed to be conducted until a "success" is achieved. In the coin-flip example, we expect that on average, it will take two flips to get a "heads".
The variance of X, var(X), is (1-p)/p2
We will use these properties in determining the mean and the variance of the solution to this riddle.
PART 2a: Finding the Mean
Let's start with finding E(Tn), the expected number of days it will take until a solution is reached, as a function of n, the number of prisoners. Let's start with the case of n = 2 just to simplify the problem.
n = 2
For simplicity's sake, let's suppose the two prisoners have selected Prisoner 1 as the principal. (It doesn't actually matter which prisoner is selected to be the principal.) On day 1, the warden selects one of the two at random. At this point, the amount of time until a new prisoner can toggle Switch 1 follows a geometric distribution with probability 1 (E(X) = 1, Var(X) = 0). This first selection is critical; there are two possible sample paths that can follow.
CASE 1: The first prisoner selected is the principal. When the principal leaves, he sees Switch 1 as down, so he knows the other prisoner has not been in yet. He can add 1 to his cumulative count. Once he returns, the amount of time until Prisoner 2 enters the room follows a geometric distribution with probability 1/2. (E(X) = 2). Then, once Prisoner 2 has been in, the principal must go back in again to receive the message through Switch 1 that Prisoner 2 has been in. The amount of time that this takes again follows a geometric distribution with probability 1/2. At that point, the principal can answer the warden's question correctly with a "Yes", and all prisoners can go free. On average, this will take 1 + 2 + 2 = 5 days.
CASE 2: The first prisoner selected is not the principal. Prisoner 2 flips Switch 1 up. The next "success" is when the principal enters the room. The amount of time that this takes again follows a geometric distribution with probability 1/2. When the principal enters, he will see Switch 1 up, and he immediately knows that all prisoners have been in since he has also been in. On average, this will take 1 + 2 = 3 days.
Now we weight these two lengths of time by the probability of each case occurring. Since selection is equally likely and there are two prisoners, each case can occur with probability 1/2. Therefore, the overall mean length of time that it will take for n = 2 prisoners to get out is 0.5*5 + 0.5*3 = 4 days. So E(T2) = 4.
Now let us consider the case of n = 3 with the same logic.
n = 3
On day 1, the amount of time until a new prisoner can toggle Switch 1 is geometric with probability 1. However, we must again condition on which prisoner was chosen first.
CASE 1: The first prisoner selected is the principal. He will add 1 to his count. After he leaves, the amount of time until a new prisoner enters the room and can toggle Switch 1 is geometric with probability 2/3. After that, the principal must be selected again; the time this takes is geometric with probability 1/3. At this point, there is one prisoner left who must toggle Switch 1 for the first time, which will occur after an amount of time defined again by a geometric distribution with probability 1/3. The principal must once again be selected, adding on another geometric with probability 1/3. Adding up the mean of all these distribution gives an expected time of 1 + 3/2 + 3 + 3 + 3 = 11.5 days.
CASE 2: The first prisoner selected is not the principal. After this first prisoner, the next time Switch 1 will be touched is when the principal enters. This happens after geom(1/3) time. At that point, the principal can add 2 to his cumulative count. After he leaves, it is another geom(1/3) until a new prisoner can turn on Switch 1. Then the principal must be called again, which occurs after geom(1/3) time. This gives an expected time of 1 + 3 + 3 + 3 = 10 days.
Now we weight these two cases. Case 1 occurs with probability 1/3, and Case 2 occurs with probability 2/3. Therefore, the overall mean length of time that it will take n = 3 prisoners to get out is (1/3)*(23/2) + (2/3)*(10) = 10.5 days.
n = 4 [Abridged, but following the same logic.]
CASE 1: (Principal first, occurs with probability 1/4)
geom(1) + geom(3/4) + geom(1/4) + geom(2/4) + geom(1/4) + geom(1/4) + geom(1/4) yields a mean of 61/3 days.
CASE 2: (Principal not first, occurs with probability 3/4)
geom(1) + geom(1/4) + geom(2/4) + geom(1/4) + geom(1/4) + geom(1/4) yields a mean of 57/3 days.
The overall mean length of time that it will take n = 4 prisoners to get out is 58/3 days, or 19.333 days.
And so we can see a pattern forming. For a general n, we can condition on whether or not the first prisoner selected is the principal.
CASE 1: (Principal first, occurs with probability 1/n)
geom(1) + geom((n-1)/n) + geom(1/n) + geom((n-2)/n) + geom(1/n) + ... + geom(2/n) + geom(1/n) + geom(1/n) + geom(1/n). The mean of the sum of these random variables can be written as follows:
CASE 2: (Principal not first, occurs with probability (n-1)/n)
geom(1) + geom(1/n) + geom((n-2)/n) + geom(1/n) + geom((n-3)/n) + geom(1/n) + ... + geom(2/n) + geom(1/n) + geom(1/n) + geom(1/n). The mean of the sum of these random variables can be written as follows:
The mean for Case 2 is written in this way to simplify the sum. In Case 2, the principal counts 2 when he first arrives as opposed to just 1 in Case 1. So Case 2 does not have the geom((n-1)/n) term, but otherwise, the means are identical. So we can simply subtract off n/(n-1).
When we weight the mean from Case 1 by 1/n and the mean from Case 2 by (n-1)/n, we get the following beautifully simple result:
We see that the mean is quadratic and increasing in n, the number of prisoners. Plotted below is the mean number of days that will pass until the prisoners can all be released, as a function of n (from 1 to 15):
PART 2b: Finding the Variance
Of course, in practice, the mean, or expected value, rarely actually occurs. In most cases, such as when n = 3, the expected value is actually not a possible value that the distribution can take, since the length of time will always be an integer number of days.
Therefore, we may be interested in knowing how "spread out" the distribution is for the time until release. Naturally, we expect the distribution to be more spread out, or more variable, as n increases. But how can we precisely calculate the variance of this strategy?
We will make extensive use of the fact that the time until escape is a sum of geometric random variables, for which we know the variance. We will also make use of the fact that each prisoner is randomly selected. What this means is that all individual trials are independent of each other (hence we can use a geometric distribution); but furthermore, we can assume that all geometric distributions in the sum are independent of each other. This is a critical assumption, as the equation
Var(X + Y) = Var(X) + Var(Y)
in general does NOT hold unless the random variables X and Y are independent of each other. Since all of our geometric random variables are independent, we can in fact use this equation to our advantage.
Variance is defined as follows:
Var(X) = E[(X - E(X))2]
Verbally, this means that the variance of X is defined as the expected squared deviation from the mean of the X. This value is squared to ensure that the deviation is represented as a positive value; it unfortunately places disproportionate weight on large deviations.
As an aside: the phrase "standard deviation" is used in common culture frequently. The standard deviation is simply the square root of the variance. By Chebyshev's Inequality, for every distribution, at least 75% of a distribution falls within 2 standard deviations of the mean, and at least 88.9% of a distribution falls within 3 standard deviations of the mean. As a general rule of thumb, if a distribution is roughly bell-shaped, then at least 95% (or more) of the distribution will be within 3 standard deviations of the mean. That is why the standard deviation is significant.
Unfortunately, in this problem, calculating the variance is not straightforward, as it again depends on which type of prisoner is selected first. We unfortunately cannot just condition on what happens first and then weight the variance of each case by the probability of each first selection; instead, we must use the formula for conditional variance and condition on the first selection.
Var(X) = E(Var(X | Y)) + Var(E(X | Y))
It is important to note that E(X | Y) is a random variable, as is Var(X | Y). Otherwise, taking the variance and mean of these values, respectively, would not make sense.
We define the random variable Y as follows:
We could have defined it the other way around, but it does not matter. What matters is what value X, or in our case, Tn, takes on when Y is 1 or 0. Fortunately, from the exercise in calculating the mean, we know what possible values this random variable can take. We will use g() to denote a geometric random variable.
We already know what E(Tn | Y) is. This random variable's possible values are the means that were calculated above. Case 1 represents E(Tn | Y = 1) and Case 2 represents E(Tn | Y = 0). Consequently, we also know E(E(Tn | Y)), which by the definition of conditional expectation, is just E(Tn). We can use these values to calculate the second part of the conditional variance formula, Var(E(Tn | Y)). This will be done by using the basic definition of variance, as defined above (expected squared deviation from the mean).
After a whole bunch of terms cancelling out, this simply reduces to:
Intuitively, this should make sense. As n grows large, the difference in possible values for E(Tn | Y) asymptotically and decreasingly approaches 1. Additionally, the probability of one of the two values occurring approaches 1. Both of these factors explain why the variance of E(Tn | Y) is decreasing in n.
Now we must calculate E(Var(Tn | Y)). This is where we can weight the variances that arise based on which prisoner is selected first. We must take the variance of the two possible outcomes for Tn. This is fairly simple by the formula for the variance of a geometric random variable, although it is a mess to write out.
I hope you're good at rearranging your fractions, because nicely enough, it simplifies to this:
Recall that we must add E(Var(Tn | Y)) and Var(E(Tn | Y)) to find Var(Tn). When we do that, the 1/(n-1) terms cancel out, and we are left with the following formula for the variance:
This result is also strikingly simple. The variance is cubic in n, implying that increasing n has a greater impact on the variance than on the mean, which intuitively makes sense. Plotted below is the variance of the time to be released, as a function of n (from 1 to 15):
Now, for you naysayers out there, I also built a simulation to model this game strategy with different numbers of n to verify if these results were correct. They are.
Anyways. I had a good nerding out about this. Hope you enjoyed this problem.
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