Two sixes in a row, fair die or loaded die?

Question

You have two dice: a fair die, and a loaded die that shows a six half the time (the other faces share the rest). You pick one at random and roll it twice. Both rolls are sixes.

What is the probability you picked the loaded die?

Accepted Answer

Each die starts with prior 1/2. Compare the chance of two sixes under each. - Fair die: (1/6)^2 = 1/36. - Loaded die: (1/2)^2 = 1/4 = 9/36. Weight by prior and likelihood (the equal priors cancel, so just compare likelihoods): P(	ext{loaded} \mid 66) = \frac{9/36}{9/36 + 1/36} = \frac{9}{10} = 0.9. Two sixes is nine times more likely from the loaded die, so belief jumps from 50\% to 90\%. Takeaway. When one hypothesis makes the observed data far more probable, the posterior odds swing by exactly that likelihood ratio, here 9 to 1.

Two sixes in a row, fair die or loaded die?

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