1995 AJHSME Problems/Problem 20
Contents
Problem
Diana and Apollo each roll a standard die obtaining a number at random from to . What is the probability that Diana's number is larger than Apollo's number?
Solution 1
Note that the probability of Diana rolling a number larger than Apollo's is the same as the probability of Apollo's being more than Diana's. If we denote this common probability , then Apollo=Diana. Now all we need to do is find Apollo=Diana. There are possibilities total, and 6 of those have Apollo=Diana, so Apollo=Diana. Going back to our first equation and solving for D, we get
Solution 2
We can use simple casework to solve this problem too. There are six cases based on Apollo's Roll. Apollo Rolls a 1: Diana could roll a , , , , or . Apollo Rolls a 2: Diana could roll a , , , or . Apollo Rolls a 3: Diana could roll a , , or . Apollo Rolls a 4: Diana could roll a or . Apollo Rolls a 5: Diana could roll a . Apollo Rolls a 6: There are no successful outcomes. The total amount of successful cases is . The total amount of possible cases is . Therefore, the probability of Diana rolling a bigger number is
See Also
1995 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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All AJHSME/AMC 8 Problems and Solutions |