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

Question

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.

Answer 1

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 $\fbox 7$.