Problem:
Diana and Apollo each roll a standard die obtaining a number at random from 1 to 6. What is the probability that Diana's number is larger than Apollo's number?
Answer Choices:
A. 31​
B. 125​
C. 94​
D. 3617​
E. 21​
Solution:
There are 6×6=36 possible outcomes of rolling the dice. Since Diana and Apollo roll the same number in 6 of these, there are 30 in which the numbers on the two dice are different. By symmetry, Diana's number is larger than Apollo's number in exactly half of these. Thus the requested probability is 3615​=125​.
OR
Let (d,a) represent "Diana rolled d and Apollo rolled a." List the 36 outcomes and mark those where d>a.
0.12,1(3,1)(4,1)5,18,1)​​1(2,2)(3,2)​(4,2)1(5,2)​(6,2)​(5,3)​(6,3)​$$(6,4)​​1(6,5)​​6(5,6)1​​
0.12,1(3,1)(4,1)5,18,1)​​1(2,2)(3,2)​(4,2)1(5,2)​(6,2)​(5,3)​(6,3)​$$(6,4)​​1(6,5)​​6(5,6)1​​
Since there are 15 marked pairs, the probability that Diana rolls a larger number than Apollo is 15/36=5/12.
Answer: B​.
The problems on this page are the property of the MAA's American Mathematics Competitions
