Expected number of draws before a certain sequence appears
Solution 1:
Hint: Try letting $E_1$ denote the expected number of more draws needed to get two $1$'s in a row if you just drew a $1$ and $E_2$ denote the expected number of more draws needed to get two $1$'s in a row if you just didn't draw a $1$. Can you get equations in terms of $E_1$ and $E_2$?