At the suggestion of a user in chat I'm posting this.
Say you have a peculiar coin. It has a head side and a tail side that are equally probable so long as no tail has been seen.
Once a tail is flipped, the coin becomes unfair, and will always show tail with probability 1 on subsequent flips.
You take a fresh coin and start flipping.
What is the expected waiting time to the first head?
My intuition is that it is infinite, since there's a non-zero probability that once you start flipping a fresh coin you'll never see a head.
Is my intuition flawed?
Basically, all that matters is the first toss. If you get $H$ then your waiting time was $1$ time unit, however if you get $T$ then you'll never see $H$. Hence, if $X$ is the number of tosses (time units) until the first $H$, its expected value is given by $$ E[X] = \frac{1}{2}\times 1 + \frac{1}{2} \times \infty = \infty. $$