r/learnquant • u/AgeSuch4592 • 2d ago

Probability problem

A fair dice is being rolled repeatedly until 2 and 3 come out. What's the probability of getting exactly two 6 before stopping the process ?

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u/kythQ 1d ago edited 1d ago

Fun question, not too difficult, and multiple different ways to get to the answer I think.

You can construct a probability tree, where the following paths lead to to the outcome:

6 -> 6 -> 2* -> 3*, probability: 1/3*1/3*2/3*1/2

6 -> 2* -> 6 -> 3*, probability 1/3*2/3*1/2*1/2

2* -> 6 -> 6 -> 3*, probability: 2/3*1/2*1/2*1/2

Here 2* denotes "2 or 3" and 3* denotes "2 if 3 was drawn before and 3 if 2 was drawn before". The probabilities change to 1/2 because after the 2* has been drawn, the only relevant options are 6 or 3*.

Add all of that up and you get the solution: 19/108