Question

Four standard dice were rolled. Then each die that did not come up six (any if) was rolled again. Following this, exactly one of the four dice showed six on its upper face. Find the probability that at least one of the four dice came up six on its initial roll.

Answer 1

: The probability of no sixes coming up is 125 216 , and in this case you lose $1. We next need the probability of exactly one 6 coming up. This can happen in one of three ways: the six can be on the first die, the second die, or the third die. For exactly one six, with it on the first die, there are 1 × 5 × 5 = 25 ways for this to happen. Similarly, there are 25 ways for the six to appear on the second die, and 25 ways for it to appear on the third die (and for exactly one 6). Thus there are 75 ways for exactly one six to appear. The value of this outcome is $1. For exactly two sixes to appear, there are 5 ways for the first two dice to be sixes, 5 ways for the first and the third dice to be sixes, and 5 ways for the last two dice to be sixes. Consequently there are 15 ways for two dice to come up six. The value of this outcome is $2. Finally, there is exactly one way for three sixes to come up, with a value of $3. Thus the expected value (for you) is 125 216 (−1) + 75 216 (1) + 15 216 (2) + 1 216 (3) = −17 216 . Hence, for every game you play, you expect to lose 7.8 cents. 12. The game “Who Wants to be a Millionaire” has $100, $200, $