Let's assume that I am playing Texas Hold'em with 5 of my friends (total 6 players). I have pockets Kings (KK). Now the flop has 3 A (AAA). I want to figure out if anyone on the board has beaten me after the flop. What would be the probability of me getting beaten?
I think the only way to beat me would be to have the final A. What would be the probability of anyone else in the game have that A?
To find the probability of one of your friends holding the last ace, we need to find the number of ways that one of them can have the ace and divide this by the total number of ways the remaining $47$ cards can be arranged. A simple way to do this would to consider those $47$ cards to be in a single deck, where the bottom $5$ pairs ($10$ cards) represent your friends' hands. The total number of ways the cards can be arranged in this deck is $47!$, and the number of arrangements with the ace in the bottom $10$ cards is $10*46!$ ($10$ possible positions for the ace and $46!$ possible arrangements of the other $46$ cards). Therefore, the probability of one of your friends holding the ace is $$\frac{10*46!}{47!} = \frac{10*46!}{47*46!} = \frac{10}{47}$$