A die's first repeated face on the third roll

Question

You roll a fair six-sided die repeatedly. What is the probability that the third roll is the first to repeat a face? (Rolls one and two are different, and roll three matches one of them.)

Accepted Answer

Two things must happen: the first two rolls are different, then the third matches one of them. Part 1, the first two differ. The first roll is anything; the second must avoid it: P(	ext{first 2 distinct}) = \frac{6}{6}\cdot\frac{5}{6} = \frac{5}{6}. Part 2, given two distinct faces are showing, the third roll matches one of those two: P(	ext{3rd matches} \mid 	ext{2 distinct}) = \frac{2}{6} = \frac{1}{3}. Multiply: P(	ext{first repeat on roll 3}) = \frac{5}{6}\cdot\frac{1}{3} = \frac{5}{18} \approx 0.278, about 27.8\%. With such a small pool (6 faces), repeats come quickly, so a first-rep…

A die's first repeated face on the third roll

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