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2007 AMC 10B Problem 22

Problem 22 of 25HarderCounting & Probability

A player chooses one of the numbers 11 through 4.4. After the choice has been made, two regular four-sided (tetrahedral) dice are rolled, with the sides of the dice numbered 11 through 4.4. If the number chosen appears on the bottom of exactly one die after it is rolled, then the player wins $1.\$1. If the number chosen appears on the bottom of both of the dice, then the player wins $2.\$2. If the number chosen does not appear on the bottom of either of the dice, the player loses $1.\$1. What is the expected return to the player, in dollars, for one roll of the dice?

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Solution

Each die shows the chosen number on the bottom with probability 14.\dfrac14. So the number appears 0,0, 1,1, or 22 times with probabilities P(0)=916, P(0)=\dfrac{9}{16},\ P(1)=616, P(1)=\dfrac{6}{16},\ P(2)=116.P(2)=\dfrac{1}{16}. The expected return is (−1)⋅916(-1)\cdot\dfrac{9}{16} +(1)⋅616+(1)\cdot\dfrac{6}{16} +(2)⋅116+(2)\cdot\dfrac{1}{16} =−9+6+216=\dfrac{-9+6+2}{16} =−116.=-\dfrac{1}{16}. Thus, the correct answer is B.

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Concepts: expected value · binomial probability · dice (probability)

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.