均匀商数

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)

英文解析

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)