Poker mathematics tutorial 1.5: playing the river

On the river you either have the winning hand or the losing hand. Besides that the pot is allready big because of the previous betting rounds. According to statistics the average pokerprofessional usually gets to see the showdown only 20 to 25% of the hands with an average winrate of 55% (9-handed).

Showdown value

So what type of hands can you expect going to the showdown 20% of the time?  These are hands like: two pair, three of a kind  a straight or a flush (and higher ofcourse).

A top pair top kicker is actually not that great of a hand on showdown (when playing on a 9-10 player table), you only go to showdown with it if you can keep the pot small enough or if your opponent is a maniac playing like 35-45% of the hands he has been dealth. If your opponent payed your bets off multiple times you can expect him to have a decent hand, which usually means two pair or higher.

Using a math problem as an example how to think about your chances on the river.

The lottery with 3 doors to choose from is a decent way to understand how pokerprofessionals think about poker. Let me explain:

You are in a gaming show and you won a chance to win a ferrari. There are 3 doors presented to you and only one of them has the ferrari behind it. You choose door number 1. The commentator walks to another door and opens it to show that there is nothing behind that door. He then asks you if you want to change your choice. Should you change your choice?

The answer is yes, you should change your choice to the other door and win the ferrari 2 out of 3 times.

Sounds crazy right? Lets explain the reasoning behind it:

You choose door 1:

1/3 chance the ferrari is behind it

2/3 chance there is nothing behind the door

Lets show this in a table. F means the ferrari is behind the door and n means nothing, there are 3 possibilities:

3 doors problem

Now one door is being removed from the game, showing that the ferrari is not behind it. This means that you actually gained more information than you had before. In the table you see that by switching you will actually win 2 out of 3 times.

Example of narrowing down the hands your opponent may have

The same idea is the case with poker to. You start the hand without any information, but during the hand you will have 4 betting rounds and these should be used to find out what your opponent may be having on the river.

Lets give a very quick and visual way of showing this.

Your opponent raises to 150 preflop from middle position he is a decent poker player (playing on a table with 9 players).  Everybody before him folds. This should give us the following handrange:

midle position

You are sitting on the big blind with AA and you decide to slowplay by just calling 100 for a pot of 325.

The flop is: flop

you decide to bet 150, your opponent raises to 450.  By doing this he is actually reducing his potential hands to:

Two pair

Three of a kind

A  pair and a flushdraw

An (over)pair and a two ended straightdraw


Two overcards and a flushdraw

Or an allready made hand.

Or a crazy bluf based on the possible fact that you bluf the flop way to often.

According to calculations by’s equilab, the aces are actually winning only 60,2% of the time in this situation. However we may know more about this player. For instance that he is loose and aggressive, but he also knows when to quit the hand.

We decide to go to the turn (pot= 1225):


You bet 600 and your opponent calls (pot=2425). You move to the river:


A blank. You decide to check and your opponent moves all in for 3000 more. What do you do?

Fold ofcourse! Allthough the river card is a blank the chances of you winning this hand are only 43% according to equilab (and the handrange he may have), which is enough to call based on pot odds (3000 in a pot of 5425 or 1,8 to 1, or 36% to break even). However based on his call after the 3rd flushcard (and 4th towards a straight) hit the board and the  all-in move on the river your opponent is actually showing great strength, which gets rid of a lot of missed hands in his range. You decide to take out most of the missed hands which gives your opponent a range of:

Inwhich case the suited hands are only hearts. The river equity calculated AFTER taking the carbage out of his handrange, gives an equity of only 25%, which means you need 3 to 1 pot odds to break even.

You decide to fold and your opponent shows:


He had a nut flushdraw and an inside straightdraw and two overcards on the flop and he decided to semibluf us with a great amount of outs to back him up. After hitting the flush on the turn he decided to trap us by just calling, hoping that we would bet the river.


What this hand shows us is that by looking at the betting patterns of the player you can actually deduce that your chances are actually way smaller then you calculate, in this case you had 0% and you should have known it. It is a bit like the 3 doors math problem. The chances change much more then you expect if you get new information during each of the betting rounds.

The methodology in this guide can also be calculated based on hand combinations your opponent may be holding on the river. This advanced method can be found on:

Calculating equity before showdown

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