Expected number of draws before drawing the same card

[Guest]

Suppose two people take turns drawing cards, each from their own pile of 100 numbered cards. They don't put back the cards.
What is the expected number of times they will have to draw before one of them draws a card the other has drawn already?

I find this quite challenging since the probability of success is not constant between draws, so it isn't a binomial distribution.

Suppose person A draws. Then the chance for person B to draw the same card in the next round is 1/100.
Suppose this doesn't happen. Then the chance for person A to draw one of B's cards is 1/99 (since he can no longer draw from 100 cards).
Next person B: change is 2/98. So as long as no one has drawn another's card, the probability seems to be something like [text]{x \choose 100-x}[/tex] at draw x.

This is how far I got. I can plot this graph: https://www.desmos.com/calculator/y3oqmcf6kk
Now I'd like to compute the expected number of draws, but I am not sure how to.

Am I doing this correctly? Any tips are welcome!