AMC10 2018 B

AMC10 2018 B · Q9

AMC10 2018 B · Q9. It mainly tests Probability (basic), Invariants.

The faces of each of 7 standard dice are labeled with the integers from 1 to 6. Let $p$ be the probability that when all 7 dice are rolled, the sum of the numbers on the top faces is 10. What other sum occurs with the same probability $p$?

有7个标准骰子，每个面标有1到6的整数。设$p$为当所有7个骰子掷出时，顶面数字之和为10的概率。哪一个其他和也以相同的概率$p$出现？

(A) 13 13

(B) 26 26

(D) 39 39

(E) 42 42

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): Without loss of generality, one can assume that the numbers on opposite faces of each die add up to 7. In other words, the 1 is opposite the 6, the 2 is opposite the 5, and the 3 is opposite the 4. (In fact, standard dice are numbered in this way.) The top faces give a sum of 10 if and only if the bottom faces give a sum of 7·7 − 10 = 39. By symmetry, the probability that the top faces give a sum of 39 is also p. The distribution of the outcomes of the dice rolls has the bell-shaped graph shown below, so no other outcome has the same probability as 10 and 39.

答案（D）：不失一般性，可以假设每个骰子相对两面的点数之和为 7。换句话说，1 的对面是 6，2 的对面是 5，3 的对面是 4。（事实上，标准骰子就是这样编号的。）顶面点数和为 10 当且仅当底面点数和为 7·7 − 10 = 39。由对称性，顶面点数和为 39 的概率也为 p。掷骰结果的分布呈现如下方所示的钟形曲线，因此不存在其他结果与 10 和 39 具有相同的概率。

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