Q:

A fair die is thrown repeatedly until we obtain the same number twice in a row. 

A fair die is thrown repeatedly until we obtain the same number twice in a row.  Compute the expected number of throws. 

For this, I found $6$ finding the transition matrix and using first step analysis. 
Is there any other way to find expected number without using markov chain?  
Thanks in advance. 

A:

Would you say that a recursive argument is the same argument as the one using markov chains?  After the first toss, say the expectation is $E$.  On the next toss you either get a double (probability $\frac 16$) or you wind up back at square one (probability $\frac 56$).   Hence $$E=\frac 16 \times 1+\frac 56 \times (E+1)\implies E=6$$
Adding back that first toss, we see that the answer is $\fbox 7$.

