A fair die rolled until two fours appear in succession.

Question

A fair die rolled until two fours appear in succession. Find the probability the experiment ends on the fifth throw.

Accepted Answer

Valid patterns ending at throw 5 with the last two being 4,4 and no earlier "44": $*\,4\,*\,4\,4$, $4\,*\,4\,4$ patterns etc. Summing the cases:

$$P = \frac{25+25+125}{6^5} = \frac{175}{6^5}$$

Solutions · 1