均匀商数

Uniform Quotient

专题: Probability / 概率
难度: L4
来源: OpenQuant

题目详情

如果 $a$ 和 $b$ 在 $[0, 1]$ 中是一致的，那么 $a/b$ 在 $[1,3]$ 中的概率是多少？

If $a$ and $b$ are uniform in $[0, 1]$ what is the probability that $a/b$ is in $[1,3]$ ?

解析

如果 $a/b$ 的商位于 $[1,3]$ 中，则 $b$ 至少为 $a/3$ ，至多为 $a$ 。因此，我们可以将概率重写为： $P(\frac{a}3 < b < a)$ 我们可以将 $a$ 的任何值代入其中以求出一般条件概率： $P(\frac{a}3 < b < a | a) = a - \frac{a}3 = \frac{2a}3$ 利用总概率定律，我们可以将所有条件概率相加来求出一般概率： $P(\frac{a}3 < b < a) = \int_0^1 P(\frac{a}3 < b < a | a) da = \int_0^1 \frac{2a}3 da = \boxed{\frac13}$

import random

quotients = []
num_iters = 10_000
for i in range(num_iters):

    a = random.uniform(0, 1)
    b = random.uniform(0, 1)

    quotients.append(a/b)

in_bounds = len([q for q in quotients if q >= 1 and q <= 3])
#expecting answer close to 1/3
print(in_bounds/num_iters)

Original Explanation

If the quotient of $a/b$ lies in $[1,3]$ , $b$ is at least $a/3$ and at most $a$ . Thus, we can re-write our probability to: $P(\frac{a}3 < b < a)$ We can substitute any value of $a$ into this to find the general conditional probability: $P(\frac{a}3 < b < a | a) = a - \frac{a}3 = \frac{2a}3$ Using the law of total probability, we can sum all of the conditional probabilities to find the general probability: $P(\frac{a}3 < b < a) = \int_0^1 P(\frac{a}3 < b < a | a) da = \int_0^1 \frac{2a}3 da = \boxed{\frac13}$

import random

quotients = []
num_iters = 10_000
for i in range(num_iters):

    a = random.uniform(0, 1)
    b = random.uniform(0, 1)

    quotients.append(a/b)

in_bounds = len([q for q in quotients if q >= 1 and q <= 3])
#expecting answer close to 1/3
print(in_bounds/num_iters)