PaC Exercise 1.10

import numpy as np

Exercise 1.10

I have a fair coin and a two-headed coin. I choose one of the two coins randomly with equal probability and flip it. Given that the flip was heads, what is the probability that I flipped the two-headed coin?

Solution: Let F denote picking the fair coin and T picking the two-headed coin, respectively. Furthermore, let H denote the event that the chosen coin shows head.

Then we have

$$ P(T | H) = \frac{P(H \cap T)}{P(H)} = \frac{P(H | T) \cdot P(H)}{P(T \cap H) + P(F \cap H)} = \frac{P(H | T) \cdot P(H)}{P(H|T) \cdot P(T) + P(H|F) \cdot P(F)} = \frac{1 \cdot \tfrac 1 2}{1 \cdot \tfrac 1 2 + \tfrac 1 2 \tfrac 1 2} = \frac 2 3. $$

We can check this by simulation.

num_samples = 100000

chosen_coin = np.random.randint(low=0, high=2, size=num_samples)  # 0 = fair, 1 = two-headed
heads = np.random.randint(low=0, high=2, size=num_samples) + chosen_coin > 0

(chosen_coin * heads).sum() / heads.sum()

0.669433509041169