The game of craps is played by rolling a pair of dice. If the total comes out to 7 or 11, the shooter wins immediately. If it comes out to 2, 3, or 12, the shooter loses immediately. If any other total shows on the first roll, the player continues to roll until either his original total comes up again, in which case he wins, or a 7 comes up, in which case he loses.

What is the probability the shooter will win?

(In reply to re(3): True Solution (Flaw in common answer.) by Charlie)

First, the odds are not the same even though the die used are rolled independently for each roll, the first roll has different odds from the second, and therefore it alters the odds for the two rolls.  Further, becuase this alteration of the odds affects the overall odds of the game and the player is precluded from exiting after the second roll in certain cases, then the odds continue to be affected.  Even though each roll imparts even odds for each number on the die, the odds for the entire set are changed.  Now, the further the rolls go, the probability approaches the limit of 50/50 for winning or losing on each subsequent roll.

If the odds of winning were the same for each roll, then it would not matter.  The problem is that the first roll imparts different odds affecting the whole set.

Second, the information is not given by anyone, but is obtained through observation.  In this case you are observing not the roll of the dice, which is irrelevant at this level, but rather the length of each game.  Knowing that the odds for a win are better with game lengths less than 12, you watch for a deviation from the normal, which in the general sense is that all run lengths are equally possible.  That said, if you notice a significant number of longer runs, then the next run length, while statistically with N=infinity is indeterminable, with N sufficiently small, you can reasonably assume is going to be less than 12, giving you a better chance of winning if you are the shooter.

Notice the main point here though, and why I said it is similar to the three doors example.  You are changing the rules (and odds) in the middle of the game.  This is why the rolls can no longer be considered independent.  And as for the mathematically proven, create a state diagram for the game.  You will see that as you add more rolls, the odds do in fact approach the limit of 50/50.  The same can be done for the three doors example.

Posted by Joshua on 2011-09-05 05:08:47