A Dice is rolled til we get two 6

[RsMath]

we roll a fair dice until we get a 6 twice in arrow
what is the expectation of numbers of rolls we have to through the dice until we get 6 twice in arrow ?
hint : it's not 36

I'll appreciate it if you help me with this question.

Thanks

 

[CRGreathouse]

We roll a fair die until we get a 6 twice in a row. What is the expected numbers of rolls until we get 6 twice in a row? Hint: it's not 36.

I'd appreciate it if you helped me with this question.

Thanks.​

On the first roll, we don't have a chance to make a double 6.

On the second roll, we can be in two cases: either the last roll was a 6, in which case we have a 1/6 chance of ending, or the last roll was not, in which case we are in the first case.

So let E be the expected number of rolls in the first case and S be the expected number of rolls in the second case. E = 1/6 * S + 5/6 * E + 1, since it takes one turn and has the specified probabilities of transitioning. This simplifies to E = S + 6.

In the second case we have S = 5/6 * E + 1, since there's a 1/6 chance of being done and a 5/6 chance of looping back, plus the turn to do either.

Now you have two equations in two variables; just solve for E.

 

[RsMath]

Thanks a lot ! very helpful, and good explanation ;)