2006 AMC 10A Problems/Problem 13

Problem

A player pays \$5 to play a game. A die is rolled. If the number on the die is odd, the game is lost. If the number on the die is even, the die is rolled again. In this case the player wins if the second number matches the first and loses otherwise. How much should the player win if the game is fair? (In a fair game the probability of winning times the amount won is what the player should pay.)

$\mathrm{(A) \ } (Error compiling LaTeX. ! Missing  inserted.)12\qquad\mathrm{(B) \ } $30\qquad\mathrm{(C) \ }$50\qquad\mathrm{(D) \ } $60\qquad\mathrm{(E) \ }$100\qquad$

Solution

There are $36$ possible combinations of 2 dice rolls.

The winning combinations are $(2,2) ; (4,4) ; (6,6)$.

Since there are $3$ winning combinations and $36$ possible combinations of dice rolls, the probability of winning is $\frac{3}{36}=\frac{1}{12}$.

Let $x$ be the amount won in a fair game.

By the definition of a fair game:

$\frac{1}{12} \cdot x = 5$.

Therefore: $x = 60 \Rightarrow D$.