Roll two fair number cubes labeled 1-6. Add the numbers. Player A wins if the sum is 5 or less. Player B wins if the sum is 9 or more. Is the game fair?

Question

Roll two fair number cubes labeled 1-6. Add the numbers. Player A wins if the sum is 5 or less. Player B wins if the sum is 9 or more. Is the game fair?

Answer 1

It may help to list all 36 of the possible outcomes, from 1-1 to 6-6 along with their sums.  Then, create a frequency distribution that looks like this:

 

  Sum     Count    Win

    2            1           A

    3            2           A

    4            3           A

    5            4           A

    6            5

    7            6

    8            5

    9            4           B

   10           3           B

   11           2           B

   12           1           B

 

Since the cubes are “fair,” the 36 outcomes have an equal probability of happening.  Player A wins 10 of 36 times and Player B wins 10 of 36 times.

 

The game is fair.