It landed heads, is it the double-headed coin?

Question

A bag holds three coins: a normal coin (heads and tails), a double-headed coin, and a double-tailed coin. You pick one at random and flip it. It lands heads.

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

Accepted Answer

Each coin has prior probability 1/3. Compare how likely a heads result is under each. - Double-headed coin: heads probability 1. - Normal coin: heads probability 1/2. - Double-tailed coin: heads probability 0. Weight each by prior times likelihood: - Double-headed and heads: 	frac13 	imes 1 = 	frac13. - Normal and heads: 	frac13 	imes 	frac12 = 	frac16. - Double-tailed and heads: 	frac13 	imes 0 = 0. The total chance of heads is 	frac13 + 	frac16 = 	frac12, and the double-headed case supplies 	frac13 of it: P(	ext{double-headed} \mid 	ext{heads}) = \frac{1/3}{1/2} = \frac{2}{3}…

It landed heads, is it the double-headed coin?

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