Three heads in a row, is it the two-headed coin?

Question

A drawer holds two coins that look identical: one is a normal fair coin, and one is a trick coin with heads on both sides. You grab one at random and flip it three times. It lands heads all three times.

What is the probability you grabbed the two-headed coin?

Accepted Answer

Both coins were equally likely to be picked, so start with prior 1/2 each. Now compare how likely three heads is under each coin. - Fair coin: three heads has probability (1/2)^3 = 1/8. - Two-headed coin: three heads is guaranteed, probability 1. Weight each coin by its prior and its likelihood: P(	ext{two-headed} \mid HHH) = \frac{	frac12 \cdot 1}{	frac12 \cdot 1 + 	frac12 \cdot 	frac18} = \frac{1}{1 + 	frac18} = \frac{8}{9} \approx 0.889. Three straight heads is eight times more likely from the trick coin than the fair one, so belief climbs from 50\% to about 89\%. Takeaway. The evide…

Three heads in a row, is it the two-headed coin?

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