# Deadly Sixes Game: Explain the apparent probability paradox to me **URL:** \ **Category:** The Game Room\ **Created:** [April 7, 2023, 4:49pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326 "2023-04-07T16:49:43Z")\ **Posts on this page:** 20\ **Page:** 5
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [April 10, 2023, 6:00pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/81 "2023-04-10T18:00:15Z")
> [@Telemark](#): > > Suppose you have an unlimited group of people, and an unlimited supply of money so you can run this experiment. Before you start, is it possible for any person to know what their odds are of surviving the game?   No, because of infinities. The best we can do is after-the-fact computation of the fraction of people who played and survived. For example, we run the game once, and everyone dies on the third dice roll. We had 11 winners, 100 losers, and an uncountable number of people who didn’t get to play. So 11/111 players won. But we cannot compute the fraction of all people who survived, because an uncountable number of people did not participate. Nor could we compute the probability that a person would play the game. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [April 10, 2023, 6:41pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/82 "2023-04-10T18:41:41Z")
> [@Pleonast](#): > > > [@Telemark](#): > > > > Suppose you have an unlimited group of people, and an unlimited supply of money so you can run this experiment. Before you start, is it possible for any person to know what their odds are of surviving the game? > > No, because of infinities. Suppose we do, in fact, have an unlimited group of people. This means infinitely many people, doesn’t it? (What else could it mean?) That means that, after the experiment is over, there will be finitely many people who have entered the room and infinitely people who have still not entered the room. Doesn’t this mean that any one person has probability 0 of entering the room (either to win or to die)? ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [April 10, 2023, 7:51pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/83 "2023-04-10T19:51:33Z")
> [@Thudlow\_Boink](#): > > That means that, after the experiment is over, there will be finitely many people who have entered the room and infinitely people who have still not entered the room. Doesn’t this mean that any one person has probability 0 of entering the room (either to win or to die)? I wouldn’t argue with it for informal purposes, but others might. Strictly speaking, zero is the limit as the number of people increases without bound, but that’s not the same as an individual person having zero probability of playing the game. You’ll get another “paradox” of “if the probability of me playing the game is zero, then how did I end up playing the game?” ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [April 10, 2023, 8:04pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/84 "2023-04-10T20:04:19Z")
> [@Pleonast](#): > > You’ll get another “paradox” of “if the probability of me playing the game is zero, then how did I end up playing the game?” Note that “probability = 0” does not mean “impossible,” in cases where the sample space is infinite. (If anyone needs a cite or explanation for this claim, see [https://www.youtube.com/watch?v=ZA4JkHKZM50](https://www.youtube.com/watch?v=ZA4JkHKZM50) or [Almost surely - Wikipedia](https://en.wikipedia.org/wiki/Almost_surely#Throwing_a_dart.)) ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [April 10, 2023, 8:20pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/85 "2023-04-10T20:20:34Z")
> [@Pleonast](#): > > No, because of infinities. The best we can do is after-the-fact computation of the fraction of people who played and survived. No, figuring out the answer does not require dealing with infinities. ---
**Author:** ![TroutMan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/troutman/32/6721_2.png) [@TroutMan](https://boards.straightdope.com/u/TroutMan)\ **Post date:** [April 10, 2023, 8:23pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/86 "2023-04-10T20:23:05Z")
> [@Little\_Nemo](#): > > No, figuring out the answer does not require dealing with infinities. Well, you might want to have a word with the OP then. > [@Little\_Nemo](#): > > For purposes of this scenario, assume there is an infinite supply of people, money, poison gas, and space in the room. ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [April 10, 2023, 8:48pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/87 "2023-04-10T20:48:19Z")
The counting numbers go to infinity. Two and three are counting numbers. But we can add two and three together and get an answer. Just because infinity is involved doesn’t mean it’s a factor. It isn’t in this case. The percentage remains ninety (to within a reasonable approximation) regardless of how many rounds the game takes. ---
**Author:** ![TroutMan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/troutman/32/6721_2.png) [@TroutMan](https://boards.straightdope.com/u/TroutMan)\ **Post date:** [April 10, 2023, 9:05pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/88 "2023-04-10T21:05:24Z")
90% of what? Since you’ll say “90% of those who played the game,” what is that number before the game starts? That’s why everyone is telling you that 90% can’t be calculated _until the game is over_. ---
**Author:** ![Jimmy\_Chitwood](https://avatars.discourse-cdn.com/v4/letter/j/f475e1/32.png) [@Jimmy\_Chitwood](https://boards.straightdope.com/u/Jimmy_Chitwood)\ **Post date:** [April 10, 2023, 9:09pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/89 "2023-04-10T21:09:23Z")
What are the odds for an individual room: infinity not a factor. What percentage of overall players at the end of the game will have lost: infinity not a factor. Why are those different questions: infinity. The thing that you are not assigning enough importance to is the fact that there is an intervening hand watching the outcomes and messing with the denominator. It’s kinda a reverse Monty Hall in the sense that you’ve got somebody cooking the books to guarantee the overall odds. Instead of opening a bunch of doors, he’s constantly inflating the number of people playing so that it always comes out right. What happens after your group has gone, if you didn’t lose, doesn’t retroactively change your odds. And if you lose, all the people who already played are part of the 10%, even though they were all the 90 when they played. ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [April 10, 2023, 9:35pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/90 "2023-04-10T21:35:45Z")
> [@TroutMan](#): > > 90% of what? > > Since you’ll say “90% of those who played the game,” what is that number before the game starts? That’s why everyone is telling you that 90% can’t be calculated _until the game is over_. Ninety percent is the answer. ---
**Author:** ![TroutMan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/troutman/32/6721_2.png) [@TroutMan](https://boards.straightdope.com/u/TroutMan)\ **Post date:** [April 10, 2023, 9:40pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/91 "2023-04-10T21:40:08Z")
The question was 90% of **what**? Can you answer that without referencing a number you know only after the game is over? ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [April 10, 2023, 10:09pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/92 "2023-04-10T22:09:48Z")
> [@TroutMan](#): > > The question was 90% of **what**? Can you answer that without referencing a number you know only after the game is over? Do you mean the total amount of players? No, you can’t answer that until the game is over and you know how many rounds actually occurred. But that was never the question I asked. I’ve been asking about the probability. ---
**Author:** ![Jas09](https://avatars.discourse-cdn.com/v4/letter/j/d07c76/32.png) [@Jas09](https://boards.straightdope.com/u/Jas09)\ **Post date:** [April 11, 2023, 12:47am UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/93 "2023-04-11T00:47:25Z")
> [@Little\_Nemo](#): > > No, you can’t answer that until the game is over and you know how many rounds actually occurred. > > But that was never the question I asked. I’ve been asking about the probability. This is critical. You need to compare your first sentence with your second one. You are asking about the probability of an event that has already happened. Probability does not work that way. You can’t ask me: “what is the probability that the coin you just flipped was heads?”. Probabilities are based on something that can vary (random variables). You can ask: “What is the probability that you survive if you are placed into a room and die if a double six is rolled?” Answer: About 97% You can ask: “What is the probability that a randomly selected individual out of a completed game of “deadly sixes” was a loser?” Answer: About 90% You cannot ask: “What is the probability you survive if you are selected to be part of the player pool for a game of deadly sixes?”. You cannot ask that question because there is no way to compute whether of not your are ever selected to go into the room. Because the denominator (the number of players in the pool) is infinite. And in fact, if you think about it, since the player pool must be infinite, the odds of any individual player (selected before the game begins) ever getting placed into a room is very small (I would guess zero in the limit, but it’s been a long time since I did formal probability calculations). ---
**Author:** ![Telemark](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/telemark/32/372_2.png) [@Telemark](https://boards.straightdope.com/u/Telemark)\ **Post date:** [April 11, 2023, 12:57am UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/94 "2023-04-11T00:57:34Z")
> [@Little\_Nemo](#): > > Which means your chances of winning is less than 10%. > A strange game. The only winning move is not to play. 90% of the eventual players will die. That’s not really a probability, it’s a rule of the game. The probability that any individual person that walks into the room will die remains 3%. You make think those two are in conflict but they are not. ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [April 11, 2023, 1:39am UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/95 "2023-04-11T01:39:28Z")
> [@Jas09](#): > > > [@Little\_Nemo](#): > > > > No, you can’t answer that until the game is over and you know how many rounds actually occurred. > > > > But that was never the question I asked. I’ve been asking about the probability. > > This is critical. You need to compare your first sentence with your second one. > > You are asking about the probability of an event that has already happened. I’m not sure what point you’re trying to make here. I was comparing two different questions in that post in order to show why they were different. But you seem to be saying they were the same question. > [@Jas09](#): > > Probability does not work that way. You can’t ask me: “what is the probability that the coin you just flipped was heads?”. Probabilities are based on something that can vary (random variables). > > You can ask: “What is the probability that you survive if you are placed into a room and die if a double six is rolled?” Answer: About 97% Agreed. > [@Jas09](#): > > You can ask: “What is the probability that a randomly selected individual out of a completed game of “deadly sixes” was a loser?” Answer: About 90% I’m not sure why you say the game is completed. The probability existed before the game began. > [@Jas09](#): > > You cannot ask: “What is the probability you survive if you are selected to be part of the player pool for a game of deadly sixes?”. You cannot ask that question because there is no way to compute whether of not your are ever selected to go into the room. This is just adding needless confusion. Mixing in people who play the game with people who don’t play the game is ridiculous. If you really can’t understand the problem because you’re confused by this point, read the post I made above where I described it from the viewpoint of somebody who is in the middle of playing the game. > [@Jas09](#): > > Because the denominator (the number of players in the pool) is infinite. > > And in fact, if you think about it, since the player pool must be infinite, the odds of any individual player (selected before the game begins) ever getting placed into a room is very small (I would guess zero in the limit, but it’s been a long time since I did formal probability calculations). No, this is not correct. As I have explained, the infinite supply of players, money, and room is not relevant to the question I asked. ---
**Author:** ![EllisDee](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ellisdee/32/4531_2.png) [@EllisDee](https://boards.straightdope.com/u/EllisDee)\ **Post date:** [April 11, 2023, 1:56am UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/96 "2023-04-11T01:56:30Z")
> [@Little\_Nemo](#): > > The probability existed before the game began. No, it didn’t. The 90% probability does not exist until the game ends. We don’t know how many people are in your room, and we don’t know how many people were in the final room. We need both of those numbers in order to calculate the 90%. * * * Let’s try a different tact. Completely unrelated to the quote and my reply to it, let’s change the rules of the dice roll. Instead of 66 losing, let’s imagine that 66 wins and everything else loses. Same deal where you add 10x the previous round to the room after each win until there’s a loss. But now instead of a 97% chance of winning, there is only a 3% chance. Or let’s say there’s a 20% chance to win. Or 60% chance to win. Or a 47% chance to win. It doesn’t matter\* what the percentage chance to win an individual round is. The answer to your core question is ~90% for every single one of them. That’s because the two percentages are completely unrelated. \*Except, of course, the chance can’t be 0% or 100%. ---
**Author:** ![Jas09](https://avatars.discourse-cdn.com/v4/letter/j/d07c76/32.png) [@Jas09](https://boards.straightdope.com/u/Jas09)\ **Post date:** [April 11, 2023, 2:06am UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/97 "2023-04-11T02:06:36Z")
> [@Little\_Nemo](#): > > As I have explained, the infinite supply of players, money, and room is not relevant to the question I asked. Well, you have asked a few questions, but if your question is: “Why are the two probabilities different?” then yes, the infinite supply of players is _the_ reason they are different. In addition to the approach @EllisDee gave you (modify the game however you like to realize why the infinite player pool is critical) you could also attempt to answer this question: What does it mean to “play the game” _before the game is played_. It seems to me it means “put myself in the player pool to be drawn into a room, if necessary”. It cannot mean “guarantee that I will enter a room” because there is no such guarantee - the first round could roll double-sixes. So, if you agree that “play the game” means “put myself into the pool of potential players”, and you agree that the pool of potential players must be infinite, it should be clear that the most likely outcome (by far) for a “player” of the game is to not ever be drawn into a room. > [@Little\_Nemo](#): > > Mixing in people who play the game with people who don’t play the game is ridiculous. I re-quoted this part, because I think this may be the root of our misunderstanding. Please define “play the game”. ---
**Author:** ![EllisDee](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ellisdee/32/4531_2.png) [@EllisDee](https://boards.straightdope.com/u/EllisDee)\ **Post date:** [April 11, 2023, 2:09am UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/98 "2023-04-11T02:09:24Z")
That’s another good angle. It’s a given that ~90% of players called into a room during the course of the game die. What we can’t know is your probability of being called into a room. ---
**Author:** ![eschrodinger](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/eschrodinger/32/122_2.png) [@eschrodinger](https://boards.straightdope.com/u/eschrodinger)\ **Post date:** [April 11, 2023, 4:45am UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/99 "2023-04-11T04:45:31Z")
> [@Telemark](#): > > 90% of the eventual players will die. That’s not really a probability, it’s a rule of the game. This is a great way of framing it. It’s just a given of the scenario that, of people who were part of a group that played the game, 90% died. That’s because the group that dies will always be 90% of the total people who went into a room and had a dice roll. That number has nothing to do with the roll. And as @EllisDee points out, you can change the chance of winning vs dying to anything, and as long as the rule continues to be that the numbers go up by x10, and the game ends with the last group dying, the proportion of players who died will always be ~90%. Because that’s just the rule of how the next group is made, plus the rule that the game ends on a loss. ---
**Author:** ![Thing.Fish](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thing.fish/32/9168_2.png) [@Thing.Fish](https://boards.straightdope.com/u/Thing.Fish)\ **Post date:** [April 12, 2023, 7:48am UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/100 "2023-04-12T07:48:33Z")
> [@Little\_Nemo](#): > > No, this is not correct. As I have explained, the infinite supply of players, money, and room is not relevant to the question I asked. The question as you describe it _only_ makes sense for an infinite number of players; more accurately, for an infinite number of _potential_ players. You seem to think that, assuming group 5 rolls double sixes, all the people who would have been in groups 6, 7, and so on weren’t “really” playing the game, but that seems illogical. If the number of potential players is infinite, then all but an infinitesimal number of them will neither win nor lose, because the game will end before they get summoned. If the number were finite, then the probability of any potential player having a given outcome could be calculated exactly _before_ the game began(though in that case there would also be a fourth possible outcome, of the number of players running out before the fatal number was rolled). Whether the number of potential players is infinite isn’t relevant to the odds of winning from the point of view of a player _during_ the game (97%) or to the odds of a particular player having won _after_ the game (about 10%). But the “infinite” angle is what makes it impossible to calculate the odds of a potential player having a given outcome _before_ the game. [Previous page](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326.md?page=4) [Next page](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326.md?page=6)