        # Probabilities in Poker

There have been several posts asking for flop odds lately. This chart puts most of the important flop odds all together.

Common odds when holding unpaired hole cards:
 flopping EXACTLY one pair by pairing a hole card flopping EXACTLY two pair by pairing a hole card AND pairing on the board flopping EXACTLY two pair by pairing EACH of your hole cards flopping EXACTLY trips by flopping two cards to one hole card flopping EXACTLY a full house, trips of one hole card and pairing the other flopping EXACTLY four of a kind, three cards to one of your hole cards 26.939% 2.02% 2.02% 1.347% 0.092% 0.01%

Common odds when holding paired hole cards:
 flopping EXACTLY two pair by pairing the board  flopping EXACTLY trips by flopping a set for your pocket pair flopping EXACTLY a full house, a set to your hole pair and pairing the board flopping EXACTLY a full house, by the board tripping up flopping EXACTLY four of a kind, two cards to your hole pair 16.163% 10.775% 0.735% 0.245% 0.245%

Common odds when holding two suited cards:
 flopping a flush (including the slight chance of a straight flush in some cases)  flopping a four flush 0.842% 10.944%

Common odds when holding connectors from 54 to JT:
 flopping a straight (including the slight chance of a straight flush in some cases) flopping an 8 out straight draw* 1.306% 10.449%

* An 8 out straight draw includes open ended straight draws and double barrelled gut shots.

# Computations

Holding unpaired cards and flopping EXACTLY one pair by matching a hole card.

I walked through that one above. You are looking for one of the 3 cards that match hole card 1 or the 3 cards that match hole card 2. So you are looking for one of 6 cards. Then you are looking for any card that DOES NOT match either of your hole cards (52 cards minus the 2 in your hand minus the 1 card that matched the first time minus the 5 cards that remain that match you hole cards = 44 cards). Then you are looking any card that DOES NOT match your hole cards AND does not match the card that just fell, because that would give you two pair (52 cards minus 2 in your hand minus the 1 card that matched the first time minus the 1 card that fell on the board minus the 8 cards that would match either your hole cards of the card that fell on the board = 40 cards.) Then divide by the total number of combinations, 50 cards * 49 cards * 48 cards. This comes to:

(6 * 44 * 40) / (50 * 49 * 48) = 8.97959%.

This is the chance of flopping one card to your hole cards ON THE FIRST CARD OF THE FLOP. There is also that same chance to flop the pairing card on the second card and the same chance to flop the pairing card on the third card

So you have 3 * 8.95959% or 26.939%

Holding unpaired cards and flopping EXACTLY a full house, trips of one hole card and pairing the other.

You are looking any of the 6 cards that will pair one of your hold cards, then you are looking any of the 2 remaining cards that will trip that hole card, then you card looking for any of the 3 cards that will pair your other hole card. Then divide by the number of combinations. That gives you:

(6 * 2 * 3 ) / (50 * 49 * 48) = 0.03061224%

That is the chance of flopping trips on the first two cards of the flop and pairing on the third. There are two other ways to flop this type of full house that have an equal chance: (1) tripping the first and third cards while pairing the second, (2) tripping the second and third cards while pairing the first.

So, you have 3 * 0.03061224 = 0.0918%.

Holding two suited cards and flopping a four flush.

You are looking for any of the 11 cards that are the same suit as the two cards that you hold. Then you are looking for any of the 10 cards that are the same suit as the first flop card. Then you are looking for any card that DOES NOT match the suit of the first two cards, there are 39. Then divide by the number of combinations. That gives you:

(11 * 10 * 39) / (50 * 49 * 48) = 3.648%

That is the chance of flopping the four flush on the first two cards of the flop. There are two other ways to flop a four flush that have an equal chance: (1) matching the suit of the first and third cards, (2) matching the suit of the second and third cards.

So, you have 3 * 3.648% = 10.944%

Straights

The straights are tricky. I prefer to think of them in a certain way and I acknowledge that not everyone may think about them the same as I. A mental model that works well for me and simplifies the math is computing the chance of EXACTLY 1 of EACH of 3 ranks falling on the flop when you do not hold any of those ranks in the hole. There are a lot of ways to compute that number. I am going to stick with the method I have used throughout the other explanations.

If you want the flop to contain ranks X, Y, Z when your hole cards do not contain X, Y, or Z then:

You are looking for 1 of the 4 X. Then you are looking for 1 of the 4 Y. Then you are looking for 1 of the 4 Z. Then divide by the number of combinations. That gives you:

(4 * 4 * 4) / (50 * 49 * 48) = 0.054422%

That is the chance of EXACTLY flopping X, then Y, then Z. There are 5 other ways of flopping X, Y, Z that all have an equal chance:

X Z Y
Y X Z
Y Z X
Z X Y
Z Y X

So, the odds of flopping EXACTLY one each of ranks X, Y, and Z when you do not hold either X, Y, or Z on the hole are:

6 * 0.054422 = 0.3265%

We are going to call that number S, and we are going to be using it a lot.

Holding connectors 54 through JT and flopping a straight.

These connectors have room on both sides to form a number of straights. Graphically, if you are holding cards AB, then you could flop the following straights:

X Y Z A B - - -
- X Y A B Z - -
- - X A B Y Z -
- - - A B X Y Z

So there are 4 sets of X, Y, Z that will give you a straight. So you have a 4 times S chance of flopping a straight.

4 * 0.3265 = 1.306%

Author: Pyroxene, source: flopturnriver.com

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