Counting fives before a four and a six

HardProbability Random Variables~11m

A fair $6$ -sided die is rolled repeatedly until both a $4$ and a $6$ have appeared (at least one of each). What is the probability that exactly two $5$ s are rolled during this process?

Show hints (2)+

Rolls of $1,2,3$ don't matter - restrict to $\{4,5,6\}$ rolls, each with probability $1/3$ .
Split the $5$ -count into phase 1 (before the first of $\{4,6\}$ ) and phase 2 (waiting for the second); the two are independent. Then sum over $F_1+F_2=2$ .

Answer

Reveal answer →

0.1759 (± 0.002)

Want the full step-by-step worked solution? It's part of Premium - along with a worked solution for every question in the bank.

Practise free Unlock solutions

Asked at: Jane Street, Citadel

Counting fives before a four and a six

Answer

Related questions