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What is the expected number of times we need to roll a die until we get three consecutive 6's?

I am trying to construct the transition matrix; however, I am not sure how also how to go from here.

Amber G.
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    Let's start with just getting to the transition matrix. The important thing is choosing the right state space: you want a dice roll to be one transition between states, you want one state to be the initial state, and you want one state to be the final state. What other state(s) do you need? – Ian Jan 19 '16 at 21:36
  • one state that I roll one consecutive 6 and the other state that I roll two consecutive 6's? – Amber G. Jan 19 '16 at 21:42
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    That's right. Now find the transition matrix; since your state space has 4 elements, it should be $4 \times 4$. (Don't worry about transitions from 3 consecutive 6's, that row won't matter.) – Ian Jan 19 '16 at 21:42
  • I am a bit confused about contrasting it, my thought of the first row of the 4x4 transition matrix is: 5/6, 1/12, 1/12, 0. Am I on the right track? – Amber G. Jan 19 '16 at 21:44
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    The 5/6 is right, but if you haven't rolled any 6's yet, then you can't get to 2 consecutive 6's in one step. You can only go back to 0 or go to 1. What's the probability of going to 1? – Ian Jan 19 '16 at 21:44
  • The probability of going to 1 is 1/6....? – Amber G. Jan 19 '16 at 21:46
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    That's right. Now continue like that. – Ian Jan 19 '16 at 21:47
  • so is my matrix just looks like P = [5/6, 1/6, 0, 0; 5/6, 0, 1/6, 0; 5/6, 0, 0, 1/6; 0, 0, 0, 1]? – Amber G. Jan 19 '16 at 21:49
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    That's right. Now you should have learned an approach based on the total expectation formula for solving your original problem. – Ian Jan 19 '16 at 21:50
  • Do we use the E(E(X|Y))? But three of them...I'm not sure I know how to do it :/ – Amber G. Jan 19 '16 at 21:59
  • That part is straightforward, and can be taught by me just as well as any book: define $\tau(x)=\inf { t : X(t)=3 \mid X(0)=x}$ and $r(x)=E[\tau(x)]$. Then the total expectation formula says $r(x)=1+\sum_{y \in S} P(x,y) r(y)$. Adjoin this to the boundary condition $r(3)=0$ and solve the resulting system of four linear equations in four unknowns. Then extract the information you actually want from the solution. – Ian Jan 19 '16 at 22:02
  • I will have a go, thank you so much for the help! Really appreciate it. – Amber G. Jan 19 '16 at 22:04
  • See here as well: http://math.stackexchange.com/questions/1487911/what-is-the-pmf-of-rolling-a-die-until-obtaining-three-consecutive-6s/1491528#1491528 –  Jan 19 '16 at 22:34

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