A Dice is rolled til we get two 6

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The discussion revolves around calculating the expected number of rolls needed to get two consecutive sixes when rolling a fair die. Participants outline the probability scenarios for the first and second rolls, emphasizing that the first roll cannot contribute to achieving two sixes in a row. The expected values are defined with equations representing the outcomes based on whether the last roll was a six or not. Through solving the equations, the expected number of rolls can be determined, which is notably not 36. The conversation highlights the mathematical reasoning behind the expected value calculation in this probability problem.
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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
 
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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.
 
Thanks a lot ! very helpful, and good explanation ;)
 
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