Paired flop

[Spanishplayer]

What are the odds to the flop coming paired???

[elitegimp]

Not considering what is in your hand, there are C(13,1)*C(4,2)*C(48,1) = 3744 different ways to deal a flop with a pair on it, and C(52,3) = 22100 different possible flops. So the probability of a paired flop is 3744/22100 = 17%

If you have an unpaired hand, then there are C(2,1)*(C(3,2))*C(47,1) = 282 different ways for you to flop trips (or a boat) and C(11,1)*C(4,2)*C(46,1) = 3036 ways the flop can pair without you having trips => 3318 flops with a pair on it. Additionally, there are C(50,3) = 19600 possible flops, so probability is 3318/19600 = 17% (this is actually 16.929%, compared to 16.941% above)

Lastly, if you have a pair in your hand, there are C(2,2)*C(48,1) = 48 ways you can flop quads (i.e. the board pairs your card), and C(12,1)*C(4,2)*C(46,1) = 3312 ways the board can pair and not have it be your quads => 3360 ways for the board to pair out of 19600 flops, so probability is 3360/19600 = 17% (17.143%)

So three possible answers:
* Given no information about cards dealt, 16.94%
* Given one hand dealt without a pair, 16.93%
* Given one hand dealt with a pair, 17.14%

Note 1: there is a 1/17 chance you are dealt a pair, and (1/17)*(3360/19600)+(16/17)*(3318/19600) = 3744/22100 so I am fairly confident in these calculations

Note 2: this does _not_ take into consideration the flop being 3 of the same card