Probability question

[MathBabe]

I always pick out the piece I am responding to, so don't think I'm arguing about something I'm not. I was not arguing with this comment:

I am arguing with this:

Just because there are three Aces on the board, you can't go back and re-calculate the probabilities of someone getting an Ace for a hole card. All you can do is eliminate a category of event.

Look at it this way. With 10 players, here's how the scenarios break down:
A: 13% probability that nobody has a hole card Ace
B: 37% probability there is one hole card Ace
C: 35% probability there are two hole cards Ace (may or may not be AA, we don't care)
D: 13% probability there are three hole card Aces
E: 2% probability there are four hole card Aces

If you see AAA on the flop, you know you are in the 50% of the time that there are none or one Aces in the hole cards. But that doesn't make the probability that there is a fourth Ace in play 20%, as you calculated - it's still 37%.

In fact, you could argue that since you've narrowed it down to those scenarios, and their relative probability is the same, there's actually a 26% chance you are in scenario A and a 74% chance you are in scenario B!

I think this is worth niggling about, because if you see three cards of a rank and you think there is a 4:1 chance of something having the fourth card, it's really more like 2:1. It may not happen often enough to affect your winnings, but over hundreds of thousands of hands, it you assume that fourth card isn't out there, you're going to be wrong more than you should.