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Texas Hold'm Odds of set on flop

AlexandreH

AlexandreH

Joined
9/5/10
Messages
397
Points
38
Let's say I have a pair in my hand, and want to know what are the odds of hitting the set (three of a kind) on the flop.
I ran my calculation and got 141/1225 = 11.5%. I have seen on multiple site a probability of 11.8%.
My calculation:
P(set on flop | pair in hand) = [2! x (48! / 46! x 2!)] / [50! / (47! x 3!)] = 141/1225

Sites that I think have it wrong are:
http://www.homepokergames.com/odds.php
http://www.learn-texas-holdem.com/questions/odds-of-improving-pocket-pairs.htm

What I think they are doing is adding the the probability I have, the probability of hitting "4 of a kind" on the flop if the hand is a pair.
 
I have a little bit of hard time understanding your equation, but I am getting the same result (11.5%).

Denominator (choosing any 3 cards out of 50 remaining cards in the deck)
Numerator (choosing any 2 cards out of 48 (different rank than what you have) cards, and 1 card out of 2 remaining card (same rank as what you have).

so. P(set on flop | pair in hand) = (2C1*48C2)/(50C3) = 2256/19600 = 11.51 %

HOWEVER, This includes 'boat' possibility, so this doesn't exactly gives you the possitibilty of a set.
In other words, the hand you have after the flop could be a "full house", if the other 2 cards in 48C2 are of the same rank.

The 11.8% probability that you see on the other forums are probabiliity of getting a set or better. In poker, there is much less merit to know the possibility of "getting a set and set only" compared to "getting a set or better".
 
To Calculate the probability of getting a "Set and Set only" you need to subtract the probability of getting a "Full House with getting one more card of the same rank as the pocket pair you have".

P(full house with one more card of the same rank as the pocket pair you have | pair in hand)
= (2C1*12C1*4C2)/(50C3) = 144/19600 = .73%

P(Set and Set only on flop | pair in hand) = (2256-144)/19600 = 2112/19600 = 10.78%
 
I think the calculation you often see is the complement of not flopping a set, that is

1 - (48/50)*(47/49)*(46/48)

So yes this would include FH/quads
 
P(not hitting your pair) = # flops not hitting / # flops
= ways to order 3 of 48 misses / ways to order 3 of 50 cards
= 48*47*46 / (50*49*48)
= 0.882449

P(hitting your pair) = 1 - P(not hitting your pair)
= 1 - 0.882449
= 0.117551

P(quads) = (# ways to pick 2 of 3 flop cards) * P(picked cards hit & rest miss)
= 3 * (# ways 2 cards both hit) * (# ways other card misses) / # flops
= 3 * (2*1) * (48) / (50*49*48)
= 0.002448980

P(exactly trips) = (# ways to pick 1 of 3 flop cards) * P(pick hits & rest miss & different)
= 3 * (# ways pick hits) * (# ways rest miss & different) / # flops
= 3 * (2) * (# first cards) * (# second cards not pairing) / (50*49*48)
= 3 * 2 * (52-4) * (52-8) / (50*49*48)
= 0.1077551

P(boat) = (# ways to pick 1 of 3 flop cards) * P(pick hits & rest miss & match)
= 3 * (# ways pick hits) * (# ways rest miss, but match) / # flops
= 3 * (2) * (# first cards) * (# second card matching first) / (50*49*48)
= 3 * 2 * (48) * (3) / (50*49*48)
= 0.007346939

Note, as above
P(hitting your pair) = P(quads) + P(exactly trips) + P(boat)
= 0.002448980 + 0.1077551 + 0.007346939
= 0.1175510
 
Next question:

Assume starting stack size of 100bb for each player
Assume the opponent has AA
Assume the opponent always bets a pot size bet (25.5bb bet) on the flop and never folds to raises (any raise)

Scenario:

We raise 3bbs preflop with 44
Opponent reraises to 12bb with his AA

Is it correct (given assumptions) to make that call preflop or not?

If so, at what starting stack size is it indifferent to call or fold preflop?

Also if calling the preflop raise is correct, at what reraise size from our opponent makes us indifferent to folding or calling?

Some quick things to think about if anyone is interested:

There is 16.5bb in the pot (1.5bb from the small and big blind) and it is 9 more bbs for us to call.
If we call the 9bb reraise preflop, the pot size will be 25.5bb and remaining stack sizes will be 88bb

bb= big blinds...if you're not familiar with poker, just make every bb $1...so we raise $3 preflop and opponent raraises to $12
 
ok. Just because I am bored...

Scooot said:
Next question:

Assume starting stack size of 100bb for each player
Assume the opponent has AA
Assume the opponent always bets a pot size bet (25.5bb bet) on the flop and never folds to raises (any raise)

Scenario:

We raise 3bbs preflop with 44
Opponent reraises to 12bb with his AA

Is it correct (given assumptions) to make that call preflop or not?

If so, at what starting stack size is it indifferent to call or fold preflop?

Also if calling the preflop raise is correct, at what reraise size from our opponent makes us indifferent to folding or calling?

Some quick things to think about if anyone is interested:

There is 16.5bb in the pot (1.5bb from the small and big blind) and it is 9 more bbs for us to call.
If we call the 9bb reraise preflop, the pot size will be 25.5bb and remaining stack sizes will be 88bb

bb= big blinds...if you're not familiar with poker, just make every bb $1...so we raise $3 preflop and opponent raraises to $12
Click to expand...

Alright, so this problem brings important poker fundamentals called "pot odds" and "implied odds". Also, we need to make a lot of assumptions. First and foremost, I will assume that everyone else has already folded or will definitely fold if i call.
Without going into detail what "pot odds" and "implied odds" are..

I am facing with two choices:
A) Fold : Result: -$3 from the raise that I made.
Risk: None
B) Call:
Three Possible scenarios after the flop
i) I don't flop a 4.
Action: I fold to the pot bet from the other guy
P(flop doesn't contain a 4) = (46C3)/(48C3) = .8777
Result: -$12 from the pre-flop raise+call
E(X_i) = .8777*(-$12) = -$10.53

ii) I flop a 4 or better without Ace hitting. + I flop two 4s and an ace (gives me 4 of a kind and gives the other guy a boat)
Action: The other guy bets the pot (22.5b) and I go all-in, and the other guy instantly calls.
P(hitting a 4 + two non-4,Ace cards) = (2C1*44C2)/(48C3) = .1094
P(hitting two 4s + a non-Ace cards) = (2C2*44C1)/(48C3) = .0025
P(hitting two 4s + an Ace) = really really small
Add all the probabilities, and P(me winning after the flop) = .1200

Turn and River
P(the other guy ends up winning, aka another ace or flush, whatever) = ~.08 (too messy to calculate)
Result: -$100 since I went all-in after the flop and lost the pot eventually.
P(My better hands holding up) = 1 - ~.08 = ~.92
Result: +$100 since I went all-in after the flop and won the pot eventually.

E(X_ii) = .12*(.08*(-$100)+.92*(+$100)) = +$10.08

iii) I do flop a 4 but the other guy hits an Ace as well! (say i'm not Phil Ivey or Jonny Chan, so I can't figure out opponent's Aces)
Action: The other guy bets the pot (22.5b) and I go all-in, and the other guy moans (acting) and then calls.
P(hitting a 4 + one Ace card) = (2C1*2C1*44C1)/(48C3) = .0102
P(hitting a 4 + two Aces): let's not count this because I will fold to this flop unless I am a fish.

Turn and River
P(I suck out and end up winning with four of a kinds or a flush or even a straight) = ~.04 (too messy to calculate)
Result: +$100
P(Oppenent's better hands holding up) = 1 - ~.04 = ~.96
Result: -$100

E(X_iii) = .0102*(.04*(+$100) + .96*(-$100)) = -$.94

So, Expected Return when calling the preflop re-raise: E(X_i) + E(X_ii) + E(X_iii) = -$10.53 + $10.08 - $0.94 = -$1.39
Risk : A LOT since the variability of your expected return is really high!!

Conclusion: If we are just comparing the expected return of these two cases:
Folding results in a loss of $3 and Calling results in a loss of $1.39, so Calling will be a right move. However, I will probably fold this hand knowing that the opponent has Aces, and that I would most likely be forced to fold after the flop and lose $12.
 
I just realized that I didn't take into account of the $1.5b from the blinds. But this shouldn't make much difference in my decision. Also, I realized that I still know what my opponent has after the flop!! So I probably know for sure if I am ahead or behind after the flop.

In real life, I always call this kind of raise because my hands might already be better than my opponent's hands and the implied odds if I hit a 4 is very high. Also, I could probably be able to figure out if I am beat even if I flop a 4.
 
AceOfSpades said:
To Calculate the probability of getting a "Set and Set only" you need to subtract the probability of getting a "Full House with getting one more card of the same rank as the pocket pair you have".

P(full house with one more card of the same rank as the pocket pair you have | pair in hand)
= (2C1*12C1*4C2)/(50C3) = 144/19600 = .73%

P(Set and Set only on flop | pair in hand) = (2256-144)/19600 = 2112/19600 = 10.78%
Click to expand...
Thank you guys. I totally missed the FH flop. Makes sense. And I agree, in poker we look for set or better, but in my case it was just an exercise of probability.
 
Scooot said:
Next question:
Click to expand...

We want to find out the odds of being ahead on the flop. Consider flops that give us trips or a boat without having an ace; if we hit quads, we can allow an ace. Let's modify your calculations for "exactly trips" and "boat" and "quads", taking into account AA and 44 are out of the deck.

P(quads) = (# ways to pick 2 of 3 flop cards) * P(picked cards hit & rest miss)
= 3 * (# ways 2 cards both hit) * (# ways other card misses) / # flops
= 3 * (2*1) * (46) / (48*47*46)
= 0.00265957447

P(exactly trips) = (# ways to pick 1 of 3 flop cards) * P(pick hits & rest miss & different & are not ace)
= 3 * (# ways pick hits) * (# ways rest miss & different & not ace) / # flops
= 3 * (2) * (# first cards) * (# second cards not pairing & not ace) / (48*47*46)
= 3 * 2 * (52-8) * (52-12) / (48*47*46)
= 0.101757632

P(boat) = (# ways to pick 1 of 3 flop cards) * P(pick hits & rest miss & match & not ace)
= 3 * (# ways pick hits) * (# ways rest miss, but match & not ace) / # flops
= 3 * (2) * (# first cards) * (# second card matching first & not ace) / (48*47*46)
= 3 * 2 * (44) * (3) / (48*47*46)
= 0.00763182239

And so

P(ahead on the flop) = P(quads) + P(exactly trips) + P(boat)
0.00265957447 + 0.101757632 + 0.00763182239
= 0.112049029

Pre-flop, we need to call a bet of 9bb. If we hit a flop where we're ahead, we're certainly going all in on the flop, for a potential gain of 100bb (neglecting rake). Now, we won't always win when we're ahead on the flop. Opponent could achieve quads on the turn and river, for instance. But we're putting our money in on the flop when we're largely ahead. To answer your questions:

Is it correct (given assumptions) to make that call preflop or not? Yes. There's a 11.2% chance of flopping ahead. Because we're calling 9% of our potential gain, which is less than 11.2%, I think we're correct in calling.

If so, at what starting stack size is it indifferent to call or fold preflop? Let x be the starting stack. We're indifferent when 9/x = .112 or when x = 9/.112 = approximately 80bb.

Also if calling the preflop raise is correct, at what reraise size from our opponent makes us indifferent to folding or calling? We're indifferent when we need to put in about 11bb pre-flop, or a reraise size of 14bb.
 
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