Different Dice

Question

Two fair dice are rolled and you are told the two faces are different. What is the probability that at least one of them is a 6?

Accepted Answer

Conditioning on "different" leaves 36 - 6 = 30 equally likely ordered outcomes. Pairs containing a 6: the 6 can sit on the first die with 5 choices for the other face, or on the second die with 5 choices, and (6,6) is excluded by the conditioning, so there is no overlap to worry about: P(textat least one 6 mid textdifferent) = frac1030 = frac13. Cross-check via the definition P(text6 and different) = frac1036, P(textdifferent) = frac3036, ratio frac1030, same thing. A detail worth noticing Unconditionally, P(textat least one 6) = frac1136 approx 0.306. Learning the dice differ raised the probability to frac1236 = frac13. The news removed six outcomes, only one of which (6,6) contained a six, so proportionally the six-containing outcomes got more weight. Tracking the direction of an update like this, before computing it, is a habit interviewers explicitly reward: it catches sign errors, and it is exactly how you sanity-check a fair-value update when a correlated instrument ticks.

Different Dice

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