expectation problem cont.

[pete2323]

Let X be a random variable representing the number of times you need to roll (including
the last roll) a fair six-sided dice until you get 4 consecutive 6's. Find E(X).
A. 125 B. 1024 C. 1554 C. 2048 D. 15447
Solution.
Note that if X = n, then the last four rolls are 6’s, the one just before was not a 6, and there
were no four consecutive 6’s on the n - 5 rolls before that.


The solution is the p(x)=n= (1/6)^4 * 5/6* P (x> n -5)





I do not understand the P x > n-5 part in this equation. Can someone explain this with an example by choosing a value for n.

[pete2323]

I just need a verbal explanation

[NoMoreExams]

I just need a verbal explanation

Search here and on AO. This problem gets discussed every few months it seems.

[pete2323]

I only see solutions posted on AO. I understand that the last 4 rolls have to be 6 so (1/6)^4 and the one before must not be a six so 5/6.



Lets assume x = 8, then one solution could be 6, 6, 6, no 6, 6,6,6,6 and another could be 6, no 6,6, no 6, 6,6,6,6

[pete2323]

Ohh I think I get it, the beginning rolls could be 1 through 6, the one before the last four cannot be six, and the last four have to be six??

[pete2323]

so if x =4 then its (1/6)^4, x = 5 its 5/6* (1/6)^4, x=6 its 1*(5/6) *(1/6)^4, x=7, its 1*1*5/6*1/6^4........ is this right??

[pete2323]

Then you just do Ex is = 1* p1 +2*p2+...... to infinity ??

[NoMoreExams]

Did you read daaave's explanation? http://www.actuarialoutpost.com/actuarial_discussion_forum/showthread.php?t=111709&highlight=consecutive

[pete2323]

I am going to read his solution again, it is very involved though..... Does my solution make sense or no?

[pete2323]

No response to my question, just what I figured from NoMoreExams

[pete2323]

1*P[1] + 2*P[2] +3*P[3] + 4*P[4]+......


I am trying to find a solution to this problem using the basic expectation formula. The first three terms would be 0. The fourth would be 4*1/6^4, and then every other term would be multiplied by 5/6 * 1/6^4. Can anyone help me sum this arithmetic series?