Four heads in a row, fair coin or biased coin?

Question

You have two coins: a fair coin, and a biased coin that lands heads $80\%$ of the time. You pick one at random and flip it four times. It comes up heads all four times.

What is the probability you picked the biased coin?

Accepted Answer

Each coin starts with prior 1/2. Compare the chance of four heads under each. - Fair coin: (1/2)^4 = 1/16 = 0.0625. - Biased coin: (0.8)^4 = 0.4096. Weight by prior and likelihood: P(	ext{biased} \mid HHHH) = \frac{0.5 \cdot 0.4096}{0.5 \cdot 0.4096 + 0.5 \cdot 0.0625} = \frac{0.4096}{0.4721} \approx 0.868 = 87\%. Four heads is far more likely from the biased coin, so the run raises belief from 50\% to about 87\%, though not as high as a two-headed coin would, since the biased coin sometimes shows tails. Takeaway. With a biased (but not certain) coin, a long head run still moves belief a lot…

Four heads in a row, fair coin or biased coin?

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