# 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:** 2
**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 7, 2023, 9:12pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/21 "2023-04-07T21:12:49Z")
> [@TroutMan](#): > > Is it being a part of the population that will maybe walk into the room, depending on what happens to the earlier groups? That can’t be determined. One hundred percent. The premise of the question is that you are playing the game. ---
**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 7, 2023, 9:15pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/22 "2023-04-07T21:15:28Z")
So look at **Pleonast** ’s post a couple above yours. Which scenario are you asking about? (And to his post, I’ll add a fourth: chances that you will win, given that you are in the game and have not yet entered the room: cannot be determined.) ---
**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 7, 2023, 9:20pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/23 "2023-04-07T21:20:30Z")
Right: neither of the probabilities given in the OP take into account the possibility that “you” might never enter the room in the first place. ---
**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 7, 2023, 9:24pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/24 "2023-04-07T21:24:06Z")
> [@Thudlow\_Boink](#): > > Right: neither of the probabilities given in the OP take into account the possibility that “you” might never enter the room in the first place. Because that’s not the question I’m asking. ---
**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 7, 2023, 9:40pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/25 "2023-04-07T21:40:48Z")
But it’s not a question; it’s a possible outcome of one of the questions you might be asking. Can you clarify which of these you are asking? 1. Chance that you will win, given that you are in the room and the dice have not yet been rolled 2. Chance that you will win, given that you have not been assigned to a group and have not yet entered the room, and the game is ongoing 3. Chance that you won, given that you entered the room and the game is still ongoing 4. Chance that you won, given that you entered the room and the game has ended ---
**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 7, 2023, 9:46pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/26 "2023-04-07T21:46:01Z")
> [@Little\_Nemo](#): > > You are playing this game. Does this mean you are subject to a specific roll of the dice OR do you mean someone in the infinite pool of people who could be chosen if it goes enough rounds? You appear to want to define the group of people subject to the roll of the dice BEFORE you actually roll the dice, and you can’t define the former before the latter. Before any rounds happen, is Person X going to be subjected to a roll? You need to answer that before you can define the odds. Anyone not subjected to a round of the game has a 0% chance of dying and there are an infinite number of them. ---
**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 7, 2023, 10:34pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/27 "2023-04-07T22:34:20Z")
> [@Pleonast](#): > > Chance of winning, given that you are currently playing the game and the dice have not yet been rolled: 35/36. > > Chance that you won, given that you were in the game and the game is still ongoing: 1. > > Chance that you won, given that you were in the game and the game has ended: limit is 1/10. More precise answers: Chance of winning, given you are queued to be in the game, and will enter the room after _n_ dice rolls: (35/36)_n_+1. Chance of winning, given you were in the game and the game is ongoing: 1. Chance of winning, given you were in the game, the game is over, and there were _n_ dice rolls: [1-9/(10_n_+1-1)]/10. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [April 7, 2023, 10:42pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/28 "2023-04-07T22:42:48Z")
> [@Little\_Nemo](#): > > I don’t feel this is true. You can, for example, determine the odds of dealing a straight flush even though theoretically you could deal yourself an infinite number of hands and never have a straight flush. Determining the probability of something happening is not the same as determining what will happen. Yes, because everything involved in that problem is finite. There is no upper bound on the number of hands you could be dealt before getting a straight flush, but the expected value of the number of hands is finite. In the problem in the OP, however, the expected number of players is not finite. ---
**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 7, 2023, 10:53pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/29 "2023-04-07T22:53:23Z")
> [@Pleonast](#): > > Chance of winning, given you are queued to be in the game, and will enter the room after _n_ dice rolls: (35/36)_n_+1. It’s important to clearly define winning in this instance. The OP said winning is surviving and getting the money, and losing is dying. There’s a third possibility, surviving and not getting the money, where you neither win nor lose. Your equation is the odds that you win, but it isn’t the same as the odds that you don’t lose. ---
**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 7, 2023, 10:59pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/30 "2023-04-07T22:59:14Z")
> [@TroutMan](#): > > There’s a third possibility, surviving and not getting the money, where you neither win nor lose. I don’t see any way in which that would happen. How do you feel that occurs? ---
**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 7, 2023, 10:59pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/31 "2023-04-07T22:59:45Z")
I am slated to go in group number 4, and group number 3 rolls double-sixes. ---
**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 7, 2023, 11:00pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/32 "2023-04-07T23:00:50Z")
Then you didn’t play the game. ---
**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 7, 2023, 11:03pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/33 "2023-04-07T23:03:04Z")
OK, that helps narrow down the question you are asking. If you aren’t “playing the game” until you step into the room, then your odds of winning are 97%. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [April 7, 2023, 11:09pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/34 "2023-04-07T23:09:02Z")
And I still say that the infinities are the source of the problem. Try setting up the problem with only a finite number of rounds, where after round, say, 10, you run out of potential players, and hence there can be no round 11. You’ll find that the odds are the same calculated either way. ---
**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 7, 2023, 11:30pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/35 "2023-04-07T23:30:08Z")
> [@Chronos](#): > > And I still say that the infinities are the source of the problem. Yep. Chance of the game having exactly _n_ dice rolls: (35/36)_n_-1/36. Which means things like the expected number of dice rolls or the expected number of players are not finite numbers. ---
**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 7, 2023, 11:39pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/36 "2023-04-07T23:39:53Z")
Infinities aren’t the only thing that is creating confusion. It absolutely is the problem with calculating odds. But **Little\_Nemo** is struggling with this not because of the difficultly in calculating the odds, but because depending on how you calculate them, you get two different answers. And that’s because they are calculating two different things that appear the same but aren’t. Given a finite number of rounds, you have the potential for three outcomes for any one individual: win money, die, or survive without winning money. Those odds can be calculated, so removing infinities solves this problem. It still leaves the problem that one case is looking at odds of winning for a single dice roll occurrence, and the other is looking at survival for a sequence of events. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [April 7, 2023, 11:51pm UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/37 "2023-04-07T23:51:41Z")
> [@TroutMan](#): > > And that’s because they are calculating two different things that appear the same but aren’t. They appear the same because they are. Again, in any finite version of the game, where the number of rounds is capped, they’ll agree. ---
**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 8, 2023, 12:01am UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/38 "2023-04-08T00:01:37Z")
No, that’s not right. These are different questions. What are the odds I will win when I walk in the room? 97%. What are the odds that I will win (i.e., get money) if it is capped at 10 rounds and I am slated for the final round? 75%, per **Pleonast** ’s equation. What are the odds that I will win if it’s capped at 10 rounds but I don’t know what round I’ll be chosen in? I’m not sure, but I know it can be calculated and it’s not either of the former two answers. And the other question the OP asks, what are the odds that I was a winner if it went 10 rounds and the participants in round 10 died? Just under 10%. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [April 8, 2023, 12:07am UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/39 "2023-04-08T00:07:28Z")
But the question the OP was asking was “Given that I’m one of the people who has played the game, what are the odds that I’m a winner?”. That’s the question that gives two different answers in the infinite version, but the same answer in the finite version. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [April 8, 2023, 12:18am UTC](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326/40 "2023-04-08T00:18:41Z")
> [@Pleonast](#): > > > [@Chronos](#): > > > > And I still say that the infinities are the source of the problem. > > Yep. > > Chance of the game having exactly _n_ dice rolls: (35/36)_n_-1/36. > > Which means things like the expected number of dice rolls or the expected number of players are not finite numbers. Let us not be hasty. If a game has a 1/36 chance each time of stopping, the expected number of rolls is only 36. [Previous page](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326.md?page=1) [Next page](https://boards.straightdope.com/t/deadly-sixes-game-explain-the-apparent-probability-paradox-to-me/982326.md?page=3)