# 2005 CEMC Gauss (Grade 7) Problems/Problem 18

## Problem

A game is said to be fair if your chance of winning is equal to your chance of losing. How many of the following games, involving tossing a regular six-sided die, are fair? $\bullet$ You win if you roll a 2 $\bullet$ You win if you roll an even number $\bullet$ You win if you roll a number less than 4 $\bullet$ You win if you roll a number divisible by 3. $\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 4$

## Solution

When a die is rolled, there are six equally likely possibilities ( $1$ through $6$). In order for the game to be fair, half of the six possibilities, or three possibilities, must be winning possibilities. In the first game, only rolling a $2$ gives a win, so this game is not fair. In the second game, rolling a $2$, $4$, or $6$ gives a win, so this game is fair. In the third game, rolling a $1$, $2$, or $3$ gives a win, so this game is fair. In the fourth game, rolling a $3$ or $6$ gives a win, so this game is not fair. Therefore, only two of the four games are fair. Thus, the answer is $C$.