去钓鱼

Gone Fishing

专题: Probability / 概率
难度: L3
来源: OpenQuant

题目详情

一只熊想要从河里捕获 $3$ 鱼。钓到 3 条鱼后，熊就离开了。当鱼来时，他有 $\frac12$ 的机会抓住它。第五条鱼没被钓到的概率是多少？

A bear wants to catch $3$ fish from a river. Upon catching 3 fish the bear leaves. When a fish comes, there is a $\frac12$ chance he'll catch it. What is the probability that the fifth fish will not be caught?

解析

只有当熊从第一条 $4$ 中捕获 $0$ 、 $1$ 或 $2$ 鱼时，它才会钓鱼第五条鱼。这是 $p = \frac12$ 的二项式： $P(F) = ({4\choose0}(\frac12)^4 + {4\choose1}(\frac12)^4 + {4\choose2}(\frac12)^4) = \frac{11}{16}$ 我们知道，当熊钓鱼时，有一半的机会会钓到第五条鱼。这对应于概率 $\frac{11}{16}*\frac{1}{2}$ 。然而，我们正在计算第五条鱼被 $not$ 捕获的概率。第五条鱼未被捕获的概率只是该事件的补充，因此我们得到： $1 - (\frac{11}{16}\cdot\frac12) = \boxed{\frac{21}{32}}$

import random

catch_fish = lambda: True if random.randint(1, 2) == 1 else False

num_fifth_fish_caught = 0
num_iters = 100_000
for i in range(num_iters):

    total_fish = 0
    for fish in range(4):

        if catch_fish():
            total_fish += 1

    if total_fish < 3:
        if catch_fish():
            num_fifth_fish_caught += 1

print((num_iters - num_fifth_fish_caught)/num_iters)

Original Explanation

The bear will only be fishing for the fifth fish when it has caught $0$ , $1$ , or $2$ fish out of the first $4$ . This is binomial with $p = \frac12$ : $P(F) = ({4\choose0}(\frac12)^4 + {4\choose1}(\frac12)^4 + {4\choose2}(\frac12)^4) = \frac{11}{16}$

We know the bear will catch the fifth fish half the time when it is fishing for it. This corresponds to the probability $\frac{11}{16}*\frac{1}{2}$ . However, we are looking to compute the probability that the fifth fish is $not$ caught. The probability the fifth fish is not caught is simply the complement of this event and so we get: $1 - (\frac{11}{16}\cdot\frac12) = \boxed{\frac{21}{32}}$

import random

catch_fish = lambda: True if random.randint(1, 2) == 1 else False

num_fifth_fish_caught = 0
num_iters = 100_000
for i in range(num_iters):

    total_fish = 0
    for fish in range(4):

        if catch_fish():
            total_fish += 1

    if total_fish < 3:
        if catch_fish():
            num_fifth_fish_caught += 1

print((num_iters - num_fifth_fish_caught)/num_iters)