Three sixes in a row, is it the loaded die?

Question

A box contains 6 dice that look identical: 5 are ordinary fair dice, and 1 is a trick die that shows a 6 on every face. You pull one out at random and roll it three times, getting 6, 6, 6.

What is the probability you grabbed the trick die?

Accepted Answer

The prior for the trick die is 	frac16 and for a fair die is 	frac56. Likelihoods of rolling three sixes: P(666\mid 	ext{trick}) = 1,\qquad P(666\mid 	ext{fair}) = \left(	frac16ight)^3 = 	frac1{216}. Form the Bayes numerator for each hypothesis: 	ext{trick: } 	frac16 	imes 1 = 	frac16, \qquad 	ext{fair: } 	frac56 	imes 	frac1{216} = 	frac{5}{1296}. Put both over a common denominator of 1296: trick = 	frac{216}{1296}, fair = 	frac{5}{1296}. So P(	ext{trick}\mid 666) = \frac{216}{216 + 5} = \frac{216}{221} \approx 0.977. Takeaway. Even a rare loaded die becomes overwhelmingl…

Three sixes in a row, is it the loaded die?

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