AMC8 1995

AMC8 1995 · Q20

AMC8 1995 · Q20. It mainly tests Basic counting (rules of product/sum), Probability (basic).

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各掷一个标准骰子，随机得到1到6的数字。Diana的数字大于Apollo的数字的概率是多少？

(A) $\frac{1}{3}$ $\frac{1}{3}$

(B) $\frac{5}{12}$ $\frac{5}{12}$

(D) $\frac{17}{36}$ $\frac{17}{36}$

(E) $\frac{1}{2}$ $\frac{1}{2}$

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): There are $6\times 6=36$ 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 $\frac{15}{36}=\frac{5}{12}$.

答案（B）：掷两枚骰子共有 $6\times 6=36$ 种可能结果。由于 Diana 和 Apollo 在其中有 6 种情况掷出相同点数，因此有 30 种情况下两枚骰子的点数不同。由对称性可知，在这些情况中，Diana 的点数恰好有一半大于 Apollo 的点数。因此所求概率为 $\frac{15}{36}=\frac{5}{12}$。

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