Two-headed coin among a hundred after one head

Question

A bag holds $100$ coins: $98$ ordinary fair coins, one coin with a head on each face, and one coin with a tail on each face. You draw a coin at random, toss it once, and see a head. What is the probability you drew the two-headed coin?

Short answer

$\tfrac1{50}$ . Heads arise with total probability $\tfrac{98}{100}\cdot\tfrac12+\tfrac1{100}\cdot1=\tfrac{50}{100}$ . The two-headed coin contributes $\tfrac1{100}$ of that, so Bayes gives $\tfrac{1/100}{50/100}=\tfrac1{50}$ .

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Lessons behind this question

Conditional Probability, Total Probability, and Bayes' Rule
Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating
Sample Spaces, Events, and the Probability Axioms

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