The flipped coin showed heads, is it the double-headed one?

Question

A box holds three coins: two are ordinary (heads on one side, tails on the other) and one is double-headed. You pick a coin at random and flip it once. It shows heads.

What is the probability you picked the double-headed coin?

Accepted Answer

Start with equal priors: each coin has probability 1/3 of being chosen. Compare how likely heads is under each type. - Double-headed coin: heads is certain, probability 1. - Ordinary coin: heads has probability 1/2. Weight each hypothesis by prior times likelihood: - Double-headed and heads: 	frac13 	imes 1 = 	frac13. - Ordinary and heads: 	frac23 	imes 	frac12 = 	frac13 (the 2/3 covers both ordinary coins together). The total chance of heads is 	frac13 + 	frac13 = 	frac23, and the double-headed case is half of it: P(	ext{double-headed} \mid 	ext{heads}) = \frac{1/3}{2/3} = \frac{…

The flipped coin showed heads, is it the double-headed one?

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