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?

You win if you roll a 2

You win if you roll an even number

You win if you roll a number less than 4

You win if you roll a number divisible by 3.

Solution

When a die is rolled, there are six equally likely possibilities ( through ). 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 gives a win, so this game is not fair. In the second game, rolling a , , or gives a win, so this game is fair. In the third game, rolling a , , or gives a win, so this game is fair. In the fourth game, rolling a or gives a win, so this game is not fair. Therefore, only two of the four games are fair. Thus, the answer is .