George and Paul play a betting game. Each chooses an integer from 1 to 20 (inclusive) at random. If the two numbers differ by more than 3, George wins the bet. Otherwise, Paul wins the bet. Calculate the probability that Paul wins the bet.

• 0.27
• 0.32
• 0.40
• 0.48
• 0.66

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: B

Let X and Y be the selected numbers. The probability Paul wins is P (| X − Y |≤ 3) . Of the 400 pairs, it is easiest to count the number of outcomes that satisfy this event:

If X = 1, then Y can be 1, 2, 3, or 4 (4 total)

If X = 2, then Y can be 1, 2, 3, 4, or 5 (5 total)

For X = 3 there are 6, and for X = 4 through 17 there are 7. For X = 18, 19, and 20 the counts are 6, 5, and 4 respectively. The total is then

4 + 5 + 6 + 14(7) + 6 + 5 + 4 = 128. 

The probability is 128/400 = 0.32.

Copyright 203. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.