This is a discussion on Probability of the board pairing? within the online poker forums, in the Learning Poker section; Please give me a correct and authoritative answer to these questions or help me figure out how to calculate them. I'm interested in the probability 


#1




Probability of the board pairing?
Please give me a correct and authoritative answer to these questions or help me figure out how to calculate them. I'm interested in the probability of the board pairing.
1. Preflop (five cards to come), what is the probability of the board pairing by the river? 2. What is the probability of the board pairing on the flop? 3. With three unpaired flop cards, what is the probability of the board pairing on the turn? By the river? 4. With four unpaired cards at the turn, what is the probability of the river card making a board pair? Thanks for your responses. Gary 
#2




BUMP.
Thirtyfive views and not one answer? If I get the answer, I'll post it. Meanwhile, somebody must know. I suppose another way of asking it is if I hold a pocket pair and have made a set on the flop, what are the odds of making a full house or better? I could figure that one. But it's not exactly the same question. 
#3




The way you'd do it; 1st card is a given. second card is 3/51=1/17 to pair. The other 16/17 of the time the third card is 6/50 to pair, multiply those #s and add to 1/17 for chance of flop being paired (or trips). Same idea for the others...If you wanted it to be for single paired boards only, you'd have to do some subtraction too.

#5




re: Poker & Probability of the board pairing?
First of all I wanna thank glworden for asking this question in the first place since it´s a very useful one when you´re playing Omaha.
I am no mathetmatical genius so I really would appreciate if someone could just give me the odds of the board pairing after a pairless flop? FLOP: I know the odds of the turn pairing is 9/49 = 18% TURN: I know the odds of the turn pairing is 12/48 = 25% Do I just add these to percentages to 43% or what? 
#6




Will depend a little on whether or not you have a pp or unpaired hole cards. But just presuming that we dont' know any cards
First card is 52/52. There is a 1/17 chance next card pairs it. If the 2nd card doesn't pair it then there is a 3/25 chance the 3rd card will pair either the first or 2nd. But if the first card hits we have to remove the % whereby we get trips on the flop. So 1/17 + ((16/17)(3/25)  (1/17)(2/25))  = .0588 + 0.109 or 17% 3 unpaired, pairing on turn: easiest way is just to do an approx. whereby you do number outs x 2 (+ 1 or 2). So you have 9 outs, so odds is about 20%. Odds from turn to river is now 12 outs so about 25%. Odds of pairing by river from unpaired flop is a little over 2:1. You can take this for granted but easiest way to figure out the odds when the amount of outs changes from turn to river is outs on turn x 2 + outs on river x 2 which in this case is 18% + 24% or 3536% ie. same odds as hitting a flush. 
#7




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If two of your Omaha hole cards pair flop cards and you're looking for one of those to pair in order to make a full house, then your odds are lower since each card only has two outs rather than three. If your pocket pair matches a flop card to make a set, then you have seven outs rather than nine to make a fullhouse or quads. And if your hole cards don't match any of the flop and you're wanting to avoid a full house suckout, the fact that you have four known hole cards makes a board pair slightly more likely since there are 9 out of 45 outs rather than 9 out of 47. If you had no hole cards and were just an observer, then you'd have to calculate 9 out of 49 outs. 
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#12




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Thank you very very much. Now I suppose if you're in the game and holding a hand of four live cards, thus reducing the number of unknowns in the deck, all of the postflop percentages quoted above would be slightly higher. I don't know how you do it, James, but you are the goto guy. Gary 
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General rule of thumb...when calculating the odds of something happening once over the course of several opportunities (such as making a pair by the river), you don't directly calculate the chance which is a pain in the ass. Instead you simply calculate the chance it won't happen at all and subtract that from one (it is by far easier that way) Quote:
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River: 1(40/49 x 36/48) = 10.61224 = 0.38776 = 33.78% Quote:
If you aren't sure where any of the numbers above came from, let me know. 
#14




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Thank you very much for the explanation. Gary 
#16




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Poker's a complex game and this is just another piece of the puzzle. GtW 
#18




Honestly, depends on hole card, whether you're holding a set or 2 pairs since turn card is a variable factor that changes the calculation, very different from other cards such as straight draw and flush draw. But i'll give you 3 situations. If you want full details of how to calculate them, read the whole comment. If you just wanna get the answer, go down to bottom of my comment.
If your hole cards didn't pair the board, then chance of board pairing is 9 outs to pair the board from the flop 19.15% chance of pairing by the turn since it's 9 cards out of 47, which is 0.1915 = 19.15% From turn, your outs increase, so it's 12 out of 46, which is 26.09% From the flop, if you're expecting to board by river, then you simple get chance of not pairing board from flopturn and turnriver and multiply it. Since chance of not pairing by turn from flop is 10019.15=80.85% and turn to river is 10026.09=73.91%, you multiply them(0.8085 x 0.7391 = 0.5976 = 56%), which gives percentage of not pairing the board from flop to river. Therefore, your chance of pairing the board if you didn't pair the flop and to see if board will pair by the river is 40.24% If you have a set, then using this precise out calculation using your calculator, you do the same thing but adjust number of outs. From flop  turn There are 7 outs. so simply 7/47 = 14.89% From turn  river Your outs increase to 10. So simply 10/46 = 21.74% Get percentage of not coming out and multiply them together to get from flopriver (0.8511 x 0.7826 = 0.6661 = 66.61%) Since chance of not pairing when holding a set is 66.61%, your chance of pairing the board from flop to river when holding a set, is 33.39%. If you flopped 2 pairs and seeking full house, then there's no adjustment in outs. You're looking for one of the 4 cards to come out by the river. So simply, from flop to turn, you calculate 4/47, which is 0.051 = 8.51%. From turn to river is 4/46, which is 0.086957 = 8.70%. To calculate making ur full house from 2 pairs, simple calculate chance of not coming out from flopturn and turnriver and then subtract that from 100 to get ur chance. So it's (0.9149 x 0.9130) = 0.8353 = 83.53%. Subtract that from 100 to get % of making ur full house from flop to river, which is 0.1647 = 16.47% Prolly got tired of reading all my long, crappy comment? I'll digest it down to simplify and summarize what I wrote. If your hole cards didnt pair the board, then chance of board pairing is Flop to turn = 19.15% Turn to river = 26.09% Flop to river = 40.24% If you have a set, then chance of pairing the board is Flop to turn = 14.89% Turn to river = 21.74% Flop to river = 33.39% If you flopped 2 pairs, chance of making your full house is Flop to turn = 8.51% Turn to river = 8.70% Flop to river = 16.47% I hope this gives full details about pairing the board and chance of making ur full house/quads. 
#19




thanks for the great answers. wonderful summar, lovesme.
So now when I'm holding the nut rainbow straight and somebody with a low set pairs the board on the river with just even money odds, I know that that is indeed a suckout. Even stupid players need to get lucky, otherwise they'd stop playing. 