Probability question: names in hats

[robvalue]

That's the sort of thing I started out with. But it doesn't take into account how the probability changes, as I mentioned above.

The first one should be 8/9, because he can't take his own name. He only actually has 9 possible picks, player 2, ..., player 10. And player 10 is excluded because that would fail the problem. So it's 8 out of 9.

The second one could be 8/9 again if player 1 took player 2's name, [1, 3, 4, ... , 10]
or 7/8 if he didn't, as his own name then gets excluded.

And so on... the probabilities are all over the place. The tree diagram is horrid. I started it, but... no! There must be a better way.