Probability question

[MathBabe]

This came up with Rogue23's four K's in the last league game, and in another post where all four of a rank were in play.

The question is, what's the probability that all four A's, for instance, are in play?

The answer depends on how many players are in the game. Let's assume a full table of 10; that gives us 20 hole cards and 5 on the table by the river. That's 25, almost half the deck.

I think my math here might be a bit dodgy, but this is what I did to calculate it:
I started by figuring out how often the first four cards off the deck are of the same rank. The first card is unimportant; the second card has a 3/51 probability of matching it; the third has 2/50, and the fourth has 1/49. That gives a combined probability of 1/20825 of the first four cards being the same.

Now for higher numbers of cards, I consider how many combinations could be four the same. If I deal five cards, for instance, there are four ways the four cards could be arranged within the five. (This is the 5 choose 4 thing again). That makes it four times more likely, or 1/5206.

For 25 cards out in play, 25 choose 4 is big - 12650. Multiplying that by the 1/20825 we get .60, making it a 60/40 split - it will happen 60% of the time, actually making it a favourite! Note that this is the probability that "four cards will be the same" - we don't care which rank they are.

To consider just one rank, say Aces, we divide by 13 again. So the odds of all four Aces being in play in a full ring game is 4.7%, or roughly 1:20.

Three-handed, by comparison, I calculate 1:62 odds of four cards the same being dealt; and only 1:818 of all four Aces (or whatever) being out.

Sorry, this is a bit hard to follow... certainly lacking in elegance!

And, of course, it's not that useful. For any given rank, you have to consider how likely it is that whoever was dealt the card stayed in with it - a bigger variable than Chance. And because you see an Ace on the flop, you know the odds that all four Aces are in play is more likely - but the odds that someone was dealt AA or a single A are the same as always, regardless of that flop Ace, since their cards were dealt first.