2026-03-24

Problem — 2026-03-24

Medium

Mediumprobability

You have two coins: one fair ( $50\%$ heads) and one biased ( $75\%$ heads). You pick a coin uniformly at random and flip it, getting heads. What is the probability you picked the fair coin?

Express as a fraction

Show solution

P(\text{fair} | H) = \frac{P(H|\text{fair}) \cdot P(\text{fair})}{P(H)} = \frac{\frac{1}{2} \cdot \frac{1}{2}}{\frac{1}{2} \cdot \frac{1}{2} + \frac{3}{4} \cdot \frac{1}{2}} = \frac{1/4}{5/8} = \frac{2}{5}