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You can roll a dice 3 times. You will be given $x where x   is the highest roll you get. You can choose to stop rolling at any time (e.g. if you roll a 6 on the first roll you can stop). What is your expected payout? 

http://math.stackexchange.com/questions/179534/the-expected-payoff-of-a-dice-game

绝妙的回答

Let's suppose we have only 1 roll. What is the expected payoff? Each roll is equally likely, so it will show 1,2,3,4,5,6 with equal probability. Thus their average of 3.5 is the expected payoff.

Now let's suppose we have 2 rolls. If on the first roll, I roll a 6, I would not continue. The next throw would only maintain my winnings of 6 (with 1/6 chance) or make me lose. Similarly, if I threw a 5 or a 4 on the first roll, I would not continue, because my expected payoff on the last throw would be a 3.5. However, if I threw a 1,2 of 3, I would take that second round. This is again because I expect to win 3.5.

So in the 2 roll game, if I roll a 4,5,6, I keep those rolls, but if I throw a 1,2,3, I reroll. Thus I have a 1/2 chance of keeping a 4,5,6, or a 1/2 chance of rerolling. Rerolling has an expected return of 3.5. As the 4,5,6 are equally likely, rolling a 4,5 or 6 has expected return 5. Thus my expected payout on 2 rolls is .5(5)+.5(3.5)=4.25.

Now we go to the 3 roll game. If I roll a 5 or 6, I keep my roll. But now, even a 4 is undesirable, because by rerolling, I'd be playing the 2 roll game, which has expected payout of 4.25. So now the expected payout is 13(5.5)+23(4.25)=4.66¯¯¯¯¯.

类似问题

1a) If you get $1 per dot on the upturned face of a rolled dice, what is the expected earnings? (Ans: $3.5)
1b) If you could have two chances to roll, and you could decide when to stop rolling (either the first roll or second roll), what would your rolling strategy be? (Ans: If you get anything less than 3.5 on the first roll, then roll again).
1c) What is your expected earning with this two roll strategy? (Ans: $4.25)