Probability of Someone Flopping a Pair as Number of Players Go Up

[colgin]

In another post I think I overestimated the probability that another player in a hand has flopped a pair assuming an unpaired board. Clearly this number goes up as the number of players seeing the goes up but how do you calculate it exactly. I know that for any one player the odds of flopping a pair is about 33%. But that does not mean that if you have three or more players someone has to flop a pair. How do you calculate this as the number of players increases. Thanks in advance.

Colgin

[The WET BEAVER]

The mathematical formula is to calculate in reverse. The odds of flopping a pair is 36% Take the number of players and multiply the odds of them not having a pair, and take the opposite number.

[Mano]

There are 49 cards other than the board, and 40 of them do not pair the board. If you have n players in, they have 2n cards - the probability that none of the 2n cards pair the board is C(40,2n)/C(49,2n). This will not take into account pocket pairs.

[scrub]

It depends what you want the number for.

The #of people taking the flop and the number of people holding a pocket pair are not independent in a real game. The board cards will also not be indepedent from the number of players taking the flop.

Calculating the probability assuming that you're playing a game with N players where everyone sees the flop without the option to fold is pretty easy, although annoying.

It's the number of ways to deal out N hands - the # of ways to deal out N hands with no PPs or pairs on the board / the number of ways to deal out N hands.

It's more annoying if you start considering the boards and getting rid of straights and flushes/draws.

scrub