Poker mathematics calculating river equity

In this guide we will discuss a way to estimate your river equity.
Let’s say thay you are capable of quickly counting the number of hands your opponent might be winning with versus the amount of hands your opponent might be losing with.

In such case you could find your  estimated iver equity quickly by calculating:

Estimated river equity=

Number of losing hands+ splitpot hands / (losing hands + winning hands+ splitpot hands)   (in which case the number of losing hands is the amount of hands your opponent may be holding that aren’t good enough to win)

Hand combinatorics for calculating river equity

For instance there are 6  ways to deal any pair with 4 cards:

(4*3)/2=6    (dividing by 2 happens because the order of the two cards don’t matter)

But how many pairs can you deal if there is already one of those cards on the board?

(3*2)/2=3

So on any flop where you have 3 decent cards you could expect there to be:

3*3=9 hands that could give your opponent a set on the flop.

Now let’s count the number of AK hands:

(8*4)/2=16   (First there are 4 A’s and 4K’s, once one of them have dropped there are only the other 4 cards left: so 8*4/2)

There are 16 ways to deal AK and 4 of those are suited hands.

But what happens if there is already any A on the flop? How many hands will be left?

The following table could make it clear:

hand combinations of AK

As you can see 4 hands have been removed with one suited hand. This leaves a total of 3*4=12 hands.

What if there drops an ace and a king on the flop, how many hands are left for your opponent to have?

In such case 3*3=9 potential AK hands he may still hold to make the two pair.

So how could all of this be of use?

Calculating river equity

You are holding:

poker hand

And the the board is:

poker board

You quickly calculate that your opponent may be winning with the following hands:

AT,KT,QT,TJs,T9s,T8s

AQ

QQ, TT, KK or AA

T6 and T5 are being neglected because of the preflop action.

66 and 55 are being neglected because of the preflop and postflop action.

Your opponents winning hands:

Two tens and four aces in the deck: 4*2=8

Two tens and 3 kings in the deck: 2*3=6

Two tens and two queens: 2*2=4

Two tens and 2 jacks (because you assume only suited TJ to be played): 2*1=2

Two tens and 4 nines (suited combinations only): 2*1=2

Two tens and 4 eights (suited combinations only):  2*1=2

Four aces and 2 queens: 4*2=8

Two queens left =1 hand

Two tens left= 1 hand

Three kings : (3*2)/2=3

Four aces: (4*3)/2=6

So there are: 8+6+4+2+2+2+8+1+1+3+6=43 hands that will win your opponent the game.

So lets say your opponent may be holding the following losing hands  :

JJ, 99,88   (6+6+6 ways of being dealth)

QJs, Q9s, (2+2 ways of being dealth)

KJ (3 kings 4 jacks =>12 ways of being dealth)

Or total air

This counts up to: 6+6+6+2+2+6+12=34 losing hands your opponent may be holding.

KQ  for a splitpot: 2Q’s and 3 K’s left=3*2=6

6 out of 83 times you will split the pot

So your river equity in this hand is: (34+6)/(34+43+6)=40/83= 48% this is exclusive his potential bluffing hands like AK.

This leads to necessary pot odds of ~ 1:1+ to call, which almost always will be the case. However it is important to note that some of the losing hands may not be played anymore on the river (the same for some winning hands usually but in this case the turn and the river are considered blanks).

The tighter your opponent the tighter the hands you calculate

Do note that 88 and 99 aren’t real hands to be counted if your opponent is a tight player, because he should fold those on postflop action. The JJ is a potential hand that could be held, however it would only give room for a small pot. In case there is an all in the JJ will be folded as well. If these 3 hands are assumed to be folded you subtract 6+6+6=18 hands from the potential losing hands your opponent may be holding.

This leads to 34-18=16 losing hands your opponent may be holding that lose him the hand.

In such case (16+6)/(16+6+43)=22/65= 0,338 river equity

This leads to 2 to 1 pot odds to be able to call.

Do note that there was no flushdraw on the flop, so I didn’t calculate the potential hands for that. If there is a flush chance there usually are about 15 suited hands that could make the flush (in case your opponent is playing a handrange of 15%). In general, the tighter your opponent is, the less hands that could make his flush. If he plays only 5% after the aggressive preflop action there are only 4 potential flushhands. In case your opponent may be holding a wide handrange of 25% he could have 28 hands that could give him the flush chance.

Practice makes perfect

As you can see this method is a great way to know your chances on the river, however it will take a lot of time to learn to quickly calculate this. In the long run you will be better of knowing this methodology, because it gives you a feeling for the value of your hand, based on similar boards you calculated in the past.

Read another post about playing the river: :

poker tutorial1.5 playing the river

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