Question

A player chooses one of the numbers $1$ through $4.$ After the choice has been made, two regular four-sided (tetrahedral) dice are rolled, with the sides of the dice numbered $1$ through $4.$ If the number chosen appears on the bottom of exactly one die after it is rolled, then the player wins $1. If the number chosen appears on the bottom of both of the dice, then the player wins $2. If the number chosen does not appear on the bottom of either of the dice, the player loses $1. What is the expected return to the player, in dollars, for one roll of the dice?

$-\dfrac{1}{8}$

$-\dfrac{1}{16}$

$0$

$\dfrac{1}{16}$

$\dfrac{1}{8}$

Solution:

Each die shows the chosen number on the bottom with probability $\dfrac14.$ So the number appears $0,1,$ or $2$ times with probabilities $P(0)=\dfrac{9}{16},\ P(1)=\dfrac{6}{16},\ P(2)=\dfrac{1}{16}.$

The expected return is $(-1)\cdot\dfrac{9}{16}+(1)\cdot\dfrac{6}{16}+(2)\cdot\dfrac{1}{16}=\dfrac{-9+6+2}{16}=-\dfrac{1}{16}.$

Thus, the correct answer is B.

2007 AMC 10B Problem 22

Solution:

Problem 22 in Other Years