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. Pre-flop (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

BUMP.
Thirty-five 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.

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.

#4 would be the easiest, you have three other cards to pair each of the four board cards, 12/48 remaning. 1/4 of the time.

...and by subtraction I meant from the 1/17 you'd have to multiply that by 48/50 to remove the chances of a trip flop, not really subtraction.

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?

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 35-36% ie. same odds as hitting a flush.

Originally Posted by thekazh

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?

Yep, I wanted to mention that I'm an Omaha player, too, and that this is very useful information whether drawing to fill a full house or protecting a made hand against a full house. I didn't mention Omaha because the stats are the same whether it's an Omaha or Hold 'em flop. Although since you know four hole cards in Omaha and two in hold 'em, the universe of the "unkown deck" is slightly different, and that would affect your calculations in a small way. But I'm looking for ballpark figures.

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.

Originally Posted by feitr

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 35-36% ie. same odds as hitting a flush.

looking at all of the math here, makes my head hurt a bit, but really makes sense when you consider what is going on. So basically there should be a board pairing 1 out of every 4 or 5 hands, seeing how their should be 3 cards to pair from. However, I would think that the more people playing would influence this number a bit (in the way that their are more chances for a community card to have its possible pairing card used up pre-flop...turn...river...). So what I will probably be using to calculate the odds would probably be one in every 6-7 hands should pair??? does that sound almost right? 14%-16% of the time??

Originally Posted by RedskinRunner325

looking at all of the math here, makes my head hurt a bit, but really makes sense when you consider what is going on. So basically there should be a board pairing 1 out of every 4 or 5 hands, seeing how their should be 3 cards to pair from. However, I would think that the more people playing would influence this number a bit (in the way that their are more chances for a community card to have its possible pairing card used up pre-flop...turn...river...). So what I will probably be using to calculate the odds would probably be one in every 6-7 hands should pair??? does that sound almost right? 14%-16% of the time??

The number of people at the table has no bearing at all.

1. 49.29%
2. 17.18%
3. 18.37% on the turn, 38.78% by the river
4. 25.00%

Here's a link you may want to check out on Tex Holdem Probabilities:

http://en.wikipedia.org/wiki/Poker_p...nds_ heads_up

Originally Posted by jamesdadeliverer

1. 49.29%
2. 17.18%
3. 18.37% on the turn, 38.78% by the river
4. 25.00%

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 post-flop percentages quoted above would be slightly higher.

I don't know how you do it, James, but you are the go-to guy.
Gary

Originally Posted by glworden

or help me figure out how to calculate them.

Calculations below. The only assumption I'm making for all of these items is that we are observing the table only and don't have a hand. Obv if we had a hand, then the % will be slightly different depending on our holding.

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)

Originally Posted by glworden

1. Pre-flop (five cards to come), what is the probability of the board pairing by the river?

1-(52/52 x 48/51 x 44/50 x 40/49 x 36/48) = 1-0.50708 = 0.49292 = 49.29%

Originally Posted by glworden

2. What is the probability of the board pairing on the flop?

1-(52/52 x 48/51 x 44/50) = 1-0.82824 = .17176 = 17.18%

Originally Posted by glworden

3. With three unpaired flop cards, what is the probability of the board pairing on the turn? By the river?

Turn: 9/49 = 18.37%
River: 1-(40/49 x 36/48) = 1-0.61224 = 0.38776 = 33.78%

Originally Posted by glworden

4. With four unpaired cards at the turn, what is the probability of the river card making a board pair?

12/48 = 25%

If you aren't sure where any of the numbers above came from, let me know.

Originally Posted by Jack Daniels

Turn: 9/49 = 18.37%
River: 1-(40/49 x 36/48) = 1-0.61224 = 0.38776 = 33.78%


If you aren't sure where any of the numbers above came from, let me know.

Was first confused that your answer differed from James, but now I just note the slight error in transcription.

Thank you very much for the explanation.
Gary

pairing the flop is usually a killer for me, I either have amazing cards and get kicked out by a weaker pair or i get my pockets cracked.

Originally Posted by darkremixx

pairing the flop is usually a killer for me, I either have amazing cards and get kicked out by a weaker pair or i get my pockets cracked.

That's the point of the question. Pairing the flop can have a huge effect on the outcome of the hand. Maybe you don't want it to pair because you have the nut flush or NF draw, or maybe you do want it to pair because you have a set drawing to a full house. But it doesn't matter what you want. We need to play what is, not what we want it to be. By knowing the likelihood of the board pairing, we can make decisions about raising, calling and folding depending on whether we're being offered proper odds for the decision.

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

Odds of the board pairing (and showing only 1 pair):

On the Flop: 4.9-1

By the Turn: 2.29-1

By the River: 1.36-1

Source: http://en.wikipedia.org/wiki/Poker_p...xas_hold_%27em)

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 flop-turn and turn-river and multiply it. Since chance of not pairing by turn from flop is 100-19.15=80.85% and turn to river is 100-26.09=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 flop-river
(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 flop-turn and turn-river 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.

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.