Three coins and a single heads

Question

A drawer holds three coins: one double-headed, one double-tailed, one fair. You pick one at random, flip it once, and see heads.

What is the probability you are holding the double-headed coin? And what is the probability the next flip is also heads?

Accepted Answer

Each coin is equally likely a priori (	frac13 each). The likelihood of heads is P(H \mid 	ext{HH}) = 1, P(H \mid 	ext{fair}) = 	frac12, P(H \mid 	ext{TT}) = 0. The total probability of heads is P(H) = 	frac13(1) + 	frac13(	frac12) + 	frac13(0) = 	frac13 + 	frac16 = 	frac12. So the posterior on the double-headed coin is P(	ext{HH} \mid H) = \frac{	frac13 	imes 1}{	frac12} = \frac{2}{3}, and the fair coin gets the remaining P(	ext{fair} \mid H) = \frac{	frac13 	imes 	frac12}{	frac12} = 	frac13 (the double-tailed coin is ruled out). Next flip. Only the double-headed coin (a…

Three coins and a single heads

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