probability of a 5-card board pairing

[DaveyDonk]

Hi. Just got done w/ a home game and they were doing all sorts of sidebets. The bet that was done most was whether the board would pair by the river w/ the 5 community cards. Does anyone know how to calculate this? I figure if they keep wanting to bet this, I may as well know how likely it is to happen and know what odds to give them on it (hell, they were even doing even odds on it for a while...). Thanks in advance.

[Tom1975]

I got
1-(48/51*44/50*40/49*36/48)=49.29%

[BoxTree]

The probability that five random cards make exactly one pair is 42.25%. This is based on the fact that there is a 42.25% chance that you'll be dealt one-pair before the draw in 5-card draw. I'm not sure of the precise math that gets to this number, but I'm sure that number is correct.

[fiskebent]

Chance that the board doesn't pair is 48*44*40*36/(51*50*49*48) or 50.71%
Chance that there's exactly one pair on the board is 13*C(4,2)*(48*44*40/6)/C(52,5) or 42.26%

[Dromar]

Wow. 42%. That's way more often than I would have guessed.

[DaveyDonk]

Damn, thought I'd be making a pretty good bet giving even odds that it would pair (me taking the side that it doesn't pair). Guess its essentially a coinflip. Thanks a bunch!