Question

某人起初有 $64$ 美分，共下注 $6$ 次，赢三次、输三次，输赢的次序随机，每次获胜和失败的概率相同。每次下注金额都是下注时剩余钱数的一半，则最终结果是：

A person starting with $64$ cents and making $6$ bets, wins three times and loses three times, the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss. If each wager is for half the money remaining at the time of the bet, then the final result is:

损失 $27$ 美分

a loss of $27$ ¢

获利 $27$ 美分

a gain of $27$ ¢

损失 $37$ 美分

a loss of $37$ ¢

既没有获利也没有损失

neither a gain nor a loss

获利或损失取决于输赢的次序

a gain or a loss depending upon the order in which the wins and losses occur

答案：C

知识点：指数过程模拟

难度评级：1470

小提示：

赢一次会把现有钱数乘以 $\frac{3}{2}$ ，输一次会把它乘以 $\frac{1}{2}$

A win multiplies the current amount by $\frac{3}{2}$ , while a loss multiplies it by $\frac{1}{2}$

大提示：

乘法交换律说明次序无关紧要

Multiplication makes the order irrelevant

解答：

三胜三负之后，剩余钱数（单位：美分）为 $\begin{aligned} 64\left(\frac32\right)^3 \left(\frac12\right)^3 &=64\cdot\frac{27}{64}\\ &=27 \end{aligned}\text{。}$ 因此无论次序如何，损失都是 $64-27=37$ 美分。

所以正确答案是 C。

After three wins and three losses, the amount, in cents, is $\begin{aligned} 64\left(\frac32\right)^3 \left(\frac12\right)^3 &=64\cdot\frac{27}{64}\\ &=27. \end{aligned}$ The loss is $64-27=37$ cents, regardless of order.

Thus, the correct answer is C.

1963 AMC 12 第 36 题

答案：C

难度评级：1470

小提示：

大提示：

解答：

其他年份的第 36 题