Probabilities

The chance of hand improvements in Texas Holdem

Assume that your pocket cards are KcTc.

The flop gives:

This is as you can see a strong hand as you have the highest pair and four cards of the same suit with high card A.

What is then the probability that you might be able to finish with the strongest hand, the nut flush?

Outs

A word that is being used when making the estimates is outs. Outs is the number of cards that are left in the pack of cards to make your hand stronger. In this case the outs we are interested in are only cards of the same suit.

The odds against improving your hand, example 1:

Calculation:

There are 13 cards of the same suit in the deck. Four of these have already been used (two in your hand and two on the table). This leaves us with 13 – 4 = 9 cards are left of that suit.

Five cards out of the 52 in the deck have already been used as you have two cards and there are three cards in the flop. In other words there are 52 – 5 = 47 cards left.

The probability that you will get your flush with the turn card is 9/47 = 0.1915 = 19.1%.

The chance that you will get your flush with the river card is 9/46 = 0.1957 = 19.6%.

BUT: If like in this case it is enough to get a flush with either the turn- or river card the procedure will be as follows:

- 9 cards results in the hand we are trying to catch

- there are 47 unseen cards at the turn and 46 unseen cards at the river.

- 38 cards at the turn card is against our flush draw: 38/47 (38 of 47)

- 37 cards at the river card is against our flush draw: 37/46 (37 of 46)

Change these figures into decimal numbers and multiply with each other:

0.81 * 0.80 = 65% missed chances

Thus the draw is successful 100% - 65% = 35% of the times.

Later you will see that this will be easier if we write this with an ”odds-formula”. Therefore I will show you how to do it.

65% missed chances against 35% equals 65:35. We want to have this in the format X:1 and therefore we divide both the numbers by 35.

This gives 1.86: 1, which is the odds AGAINST your draw. This means that for every time you succeed you will fail 1,86 times.

Please don't panic! There are charts to use for this with all the numbers already calculated!

Assume that we instead would have had two suited cards in the hand and one card of the same suit in the flop, thus we would have to hit both the turn- and river card.

The odds against improving your hand, example 2:

Calculation:

Multiply the probabilities:

(10/47) * (9/46) = 0,0416 = 4.2% probability

Expressed as odds:

As 95.8% is against and 4.2% is for a successful draw the odds will be 95.8 : 4.2

We divide both the numbers by 4.2 and thus we get the odds in the much nicer format 22.8 :1.

This means you will succeed once every 23.8th hand. In other words - now you know why the other players are inclined to call you river rat when you are drawing for a flush  although you only have three cards to a flush on the flop.