One hundred percent. The premise of the question is that you are playing the game.
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.)
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.
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?
- Chance that you will win, given that you are in the room and the dice have not yet been rolled
- 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
- Chance that you won, given that you entered the room and the game is still ongoing
- Chance that you won, given that you entered the room and the game has ended
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.
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/(10n+1-1)]/10.
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.
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.
I don’t see any way in which that would happen. How do you feel that occurs?
I am slated to go in group number 4, and group number 3 rolls double-sixes.
Then you didn’t play the game.
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%.
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.
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.
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.
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.
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%.
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.
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.