06/07/2020 (2011: National Sprint Round, Problem 21)

Q: Bob and Tom each roll two standard six-sided dice. What is the probability that the product of their individual sums is greater than 100? Express your answer as a common fraction.

A: 35/1296

The range of sums that they can get are 2 (1 + 1) to 12 (6 + 6). We list out all the possible products of two sums that are greater than 100 (The first value is Bob and the second value is Tom):

{9, 12}

{10, 11} {10, 12}

{11, 10} {11, 11} {11, 12}

{12, 9} {12, 10} {12, 11} {12, 12}

The probability of getting a 9 is 1/9. The probability of getting a 10 is 1/12. The probability of getting a 11 is 1/18. The probability of getting a 12 is 1/36.

That means the probability of getting the first outcome ({9, 12}) is 1/9 * 1/36 = 1/324

The probability of getting the second outcome ({10, 11}) is 1/12 * 1/18 = 1/216

The probability of getting the third outcome ({10, 12}) is 1/12 * 1/36 = 1/432

The probability of getting the third outcome ({11, 10}) is 1/18 * 1/12 = 1/216

The probability of getting the third outcome ({11, 11}) is 1/18 * 1/18 = 1/324

The probability of getting the third outcome ({11, 12}) is 1/18 * 1/36 = 1/648

The probability of getting the third outcome ({12, 9}) is 1/36 * 1/9 = 1/324

The probability of getting the third outcome ({12, 10}) is 1/36 * 1/12 = 1/432

The probability of getting the third outcome ({12, 11}) is 1/36 * 1/18 = 1/648

The probability of getting the third outcome ({12, 12}) is 1/36 * 1/36 = 1/1296

If you add them all up you get 35/1296.