Third head from a coin chosen among three

Question

You have three coins: one lands heads every time, one lands tails every time, and one is fair. You choose one at random and toss it twice, getting two heads. If you toss that same coin once more, what is the probability of another head?

Short answer

$9/10$ . The two heads have likelihood 1, 0 and $\tfrac14$ for the three coins, so the posterior is $\tfrac45$ on the two-headed coin and $\tfrac15$ on the fair one. The next head then has probability $\tfrac45\cdot1+\tfrac15\cdot\tfrac12=\tfrac{9}{10}$ .

Adapted from OpenQuant; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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