2020 AMC 12A 第 23 题

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23.

Jason 掷三枚公平的标准六面骰。然后他查看掷出的点数,并选择一个骰子子集(可以为空,也可以是全部三枚)重新掷。重新掷后,当且仅当三枚骰子朝上点数之和恰好为 77 时,他获胜。Jason 总是采取最优策略来最大化获胜概率。他选择恰好重新掷两枚骰子的概率是多少?

Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly 7.7. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?

736\dfrac{7}{36}

524\dfrac{5}{24}

29\dfrac{2}{9}

1772\dfrac{17}{72}

14\dfrac{1}{4}

答案:A
知识点:骰子(概率)最优化分类讨论
难度评级:2270
解答:

重掷一枚骰子、保留点数和为 s,s, 的两枚骰子时,如果 s6s \le 6,获胜概率为 16\tfrac16,否则为 00。重掷两枚骰子、保留点数为 v,v, 的一枚骰子时,获胜概率等于两枚骰子点数和为 7v,7 - v, 的方式数除以 36;36;vv 最小时,这个概率最大。对 v=1,2,3,v=1,2,3,这些概率分别是 536,436,336,\tfrac5{36},\tfrac4{36},\tfrac3{36},都大于重掷全部骰子的概率 15216=572;\tfrac{15}{216}=\tfrac{5}{72};v4,v\ge4,重掷全部骰子更好。

恰好重掷两枚骰子严格最优,当且仅当最小的两枚骰子点数和至少为 77(所以重掷一枚不能达到 77),同时最小骰子的点数是 1,2,1, 2,33(所以保留它优于重掷全部三枚)。

将结果排序为 uvw.u\le v\le w.u=1,u=1, 时,只有 (1,6,6),(1,6,6), 一种,排列数为 33。当 u=2,u=2, 时,(2,5,5),(2,5,6),(2,6,6)(2,5,5),(2,5,6),(2,6,6) 的排列数之和为 3+6+3=123+6+3=12。当 u=3,u=3, 时,从 {4,5,6};\{4,5,6\}; 中可重复地选 v,wv,w,六个三元组的排列数之和为 3+6+6+3+6+3=273+6+6+3+6+3=27。因此在 216,216, 个有序结果中,有 3+12+27=423+12+27=42 个满足条件,概率为 42216=736.\dfrac{42}{216} = \dfrac{7}{36}.

所以 A 是正确答案。

Rerolling one die, keeping two dice that sum to s,s, wins with probability 16\tfrac16 when s6s \le 6 and 00 otherwise. Rerolling two dice, keeping a die of value v,v, wins with probability equal to the number of ways two dice sum to 7v,7 - v, over 36;36; this is largest when vv is smallest. For v=1,2,3,v=1,2,3, these probabilities are 536,436,336,\tfrac5{36},\tfrac4{36},\tfrac3{36}, all greater than the reroll-all probability 15216=572;\tfrac{15}{216}=\tfrac{5}{72}; for v4,v\ge4, rerolling all is better.

Rerolling exactly two dice is strictly best precisely when the two smallest dice sum to at least 77 (so rerolling one cannot reach 77) while the smallest die is 1,2,1, 2, or 33 (so keeping it beats rerolling all three).

Sort the roll as uvw.u\le v\le w. If u=1,u=1, the only possibility is (1,6,6),(1,6,6), with 33 orderings. If u=2,u=2, the possibilities (2,5,5),(2,5,6),(2,6,6)(2,5,5),(2,5,6),(2,6,6) have 3+6+3=123+6+3=12 orderings. If u=3,u=3, choose v,wv,w with repetition from {4,5,6};\{4,5,6\}; the six resulting triples have 3+6+6+3+6+3=273+6+6+3+6+3=27 orderings. Thus there are 3+12+27=423+12+27=42 qualifying ordered rolls out of 216,216, a probability of 42216=736.\dfrac{42}{216} = \dfrac{7}{36}.

Thus, A is the correct answer.

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