I Expected number of random variables that must be observed

[WMDhamnekar]

TL;DR

Expected number of random variables that must be observed before any specific sequence.

In my opinion, answer to (a) is $\mathbb{E} [N] = p^{-4}q^{-3} + p^{-2}q^{-1} + 2p^{-1}$
In answer to (b), X_N is wrong. It should be X_N=p^-4q^-3 - p^-3 q^-2- p^-2 q^-1 - p^-1. This might be a typographical error.
Is my answer to (a) correct?

 

[haruspex]

In my opinion, answer to (a) is $\mathbb{E} [N] = p^{-4}q^{-3} + p^{-2}q^{-1} + 2p^{-1}$

Please explain your reasoning.

For b) I agree with you.

 

[WMDhamnekar]

Please explain your reasoning.

For b) I agree with you.

Answer to (a) given by author is correct. My answer is wrong. Thanks for bringing my error to my notice.

 

[haruspex]

Answer to (a) given by author is correct. My answer is wrong. Thanks for bringing my error to my notice.

You are welcome.
I had never seen this method before. It's brilliant- thanks for posting.