AMC 8 1995 Problem 20
AMC8 1995 年第 20 题
概率★★★☆☆
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?
Diana 和 Apollo 各自掷一个标准骰子，随机得到 1 到 6 中的一个数。Diana 的数大于 Apollo 的数的概率是多少？
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
B
解析 · Solution
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 . Now all we need to do is find . There are 6(6)=36 possibilities total, and 6 of those have Apollo=Diana, so . Going back to our first equation and solving for D, we get
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 2, 3, 4, 5, or 6. Apollo Rolls a 2: Diana could roll a 3, 4, 5, or 6. Apollo Rolls a 3: Diana could roll a 4, 5, or 6. Apollo Rolls a 4: Diana could roll a 5 or 6. Apollo Rolls a 5: Diana could roll a 6. Apollo Rolls a 6: There are no successful outcomes. The total amount of successful cases is 5+4+3+2+1 = 15. The total amount of possible cases is 6(6) = 36. Therefore, the probability of Diana rolling a bigger number is
Define a "win" for Player 1 as rolling a greater number than Player 2. Because potential ties make up of all 36 combinations, Apollo and Diana "split" the wins in the remaining . By symmetry, each person wins half of these, so the probability Diana wins is
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