Spotting a two-headed coin after three heads

Question

A bag holds ten coins: nine are fair and one has heads on both sides. You draw one coin at random, toss it three times, and see three heads. What is the probability you drew the two-headed coin?

Short answer

$8/17$ . By Bayes, $\dfrac{\tfrac1{10}\cdot1}{\tfrac1{10}\cdot1+\tfrac9{10}\cdot\tfrac18}=\dfrac{0.1}{0.2125}=\tfrac{8}{17}\approx0.47$ .

Adapted from Joshi, Denson & Downes, Quant Job Interview Questions and Answers, Q3.10, p.81; question and answer rewritten by SelfMadeQuant.com.

See the hint and the full worked solution in the question bank (free account).

Open the worked solution

Lessons behind this question

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

Start with a free lesson: Probability Fundamentals is part of a free topic. Create a free account, no card needed, to read it in full with worked solutions.

Read it free