Markov chain return time to a bad bagel

Bagels come off a production line one after another, each either good or bad. After a good bagel, the next is good with probability $\tfrac25$; after a bad bagel, the next is bad with probability $\tfrac35$. The current bagel is bad. What is the expected number of further bagels that come off the line up to and including the next bad one?

$5/3$. With probability $\tfrac35$ the very next bagel is bad. Otherwise it is good, and from a good bagel each step turns bad with probability $\tfrac35$, a geometric wait of mean $\tfrac53$. So the answer is $1+\tfrac25\cdot\tfrac53=\tfrac53$ (equivalently, $1/\pi_{\text{bad}}$ with stationary probability $\pi_{\text{bad}}=\tfrac35$).