prob of flopping a set from pp

Hello,

I'm fairly new and am not sure why my calculation doesn't line up with several webpages I saw. It seems to me that if you have a pocket pair, then the probability of flopping a set is 1 - probability of not getting either of your 2 outs 3 times in a row, out of the remaining 50, 49 and 48 cards.

1 - (48/50*47/49*46/48) = 11.8%

The webpages I looked at said it's supposed to be 10.8% though.
http://www.homepokergames.com/odds.php

If anything, it seems like it should be slightly higher because I am not considering the case of quads. What am I doing wrong?

Also does anyone know offhand the probability of:
a) making at least a pair by the river, when you know what was dealt
b) making at least a pair by the river, when starting without a pocket pair

thanks,
John

I've always heard that the odds to hit a set is *roughly* 1-in-8, which has always been general enough for me, limp in or dont worry too much about it (unless its JJ or higher.)

Perhaps they didn't work quads into the %?

To make at least a pair by the river assuming you didn't pair either of your hole cards on the flop:

chance at turn - 24.1%
chance at river - 13%

Not sure about the flop %

pocket pair getting a set by the river: 12% very close to 1/8
pocket pair getting quad by the river: 1%

capt_blunder:

Your calculation agrees with them.

They broke their calculation out by probability of getting a set, quads, and a FH. Adding all of theirs together gets the same probability that you got.

Here's an easy way to calculate this -- open an Excel spreadsheet and use the COMBIN function:

The total number of combinations of flops is COMBIN(50,3) (read 50 choose 3) or 19600. For your calculation there are 2 cards that create a set. You are interested in the number of combinations that will give a set. That is given by multiplying the number of cards of interest, COMBIN(2,1), by the number of cards that aren't interesting, COMBIN(48,2). That gives a total of 1128 combinations that gives you exactly a set. Dividing by the total number of possible hands gives a probability of .1152 or 11.52% chance of getting exactly a set.

Quads are done the same way:

Combin(2,2)*Combin(48,1)/Combin(50,3) = .002449 or .2449% chance of quads when holding a PP.

Adding the 2 numbers gives 0.11755, which is your answer when rounded.

What the web page where you got your answers did is break out the set, quad, and FH probabilities (FH probabilities are a little different, in that it can be made 2 ways -- 1 of your pair + another pair or trips on the flop. I can break things down further if you're interested). Your answer has any of the above in one number.

[Edit: Last line should read: Your answer has the set, quad, and the version of FH where 1 of your PP + another pair on the flop in it. The FH where you get a different trip on the flop is not in your calculations.]

Flopping the set is probably the most important number as I start to lose faith in low pocket pairs on the turn and the river.

Originally Posted by Bucko

capt_blunder:

Your calculation agrees with them.

They broke their calculation out by probability of getting a set, quads, and a FH. Adding all of theirs together gets the same probability that you got.

Here's an easy way to calculate this -- open an Excel spreadsheet and use the COMBIN function:

The total number of combinations of flops is COMBIN(50,3) (read 50 choose 3) or 19600. For your calculation there are 2 cards that create a set. You are interested in the number of combinations that will give a set. That is given by multiplying the number of cards of interest, COMBIN(2,1), by the number of cards that aren't interesting, COMBIN(48,2). That gives a total of 1128 combinations that gives you exactly a set. Dividing by the total number of possible hands gives a probability of .1152 or 11.52% chance of getting exactly a set.

Quads are done the same way:

Combin(2,2)*Combin(48,1)/Combin(50,3) = .002449 or .2449% chance of quads when holding a PP.

Adding the 2 numbers gives 0.11755, which is your answer when rounded.

What the web page where you got your answers did is break out the set, quad, and FH probabilities (FH probabilities are a little different, in that it can be made 2 ways -- 1 of your pair + another pair or trips on the flop. I can break things down further if you're interested). Your answer has any of the above in one number.

[Edit: Last line should read: Your answer has the set, quad, and the version of FH where 1 of your PP + another pair on the flop in it. The FH where you get a different trip on the flop is not in your calculations.]

Why would you need to use excel?

Just use any basic scientific calculator with basic statisical functions (nCr, nPr, n!).

If you have the n! function but not the others, recall the defintion of the combination function:

nCr = n! / (n-r)! r!

I tend to use Excel because most people have it and it's easier to use and explain than the basic combinatory expression. The COMBIN function is just a repackaging of the combinatory function you showed in the last line.

pocket pair getting a set by the river: 12% very close to 1/8

Wrong. PP hitting set on flop is about 12% or 1/8. PP hitting set by river is around 20% or 1/5.

pocket pair getting a set by the river: 12% very close to 1/8

Wrong. PP hitting set on flop is about 12% or 1/8. PP hitting set by river is around 20% or 1/5. You need to know percentages for pot odd purposes and you're making it way too complicated for real time decisions. All you need to do is calculate how many outs you have, multiply that number by 2, and then multiply it by how many cards are yet to come (Flop = 3, Turn = 1, River =1, Turn & River = 2). So with the pocket pair example you used, you have 2 outs to make your set. The odds of hitting it on the flop are 2 outs x 2 x 3 cards coming on flop = 12% (1/8). The odds of making a set by the river are 2 outs x 2 x 5 cards (flop,turn,river) = 20% (1/5). Grant is these number aren't exact, but they're pretty damn close and more than sufficient to calculate pot odds. Do it this way and you can calculate percentages on the fly and it gives you round numbers that are easy to deal with instead of decimals like 11.873645%.