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The Math of Poker

Amateurs, beginners and people who are absolutely clueless about poker may call it gambling or a ‘game of chance’. But professionals know that there is a huge element of mathematics involved, particularly probability. This article briefly looks into the math of poker. The first concept to learn in the math of poker, is the concept of pot odds.

Pot odds are basically the ratio between the size of the pot and the size of the bet which has been put forward by another player. For example, if there is 500 rupees in the pot, and you are facing a bet of 250 rupees, then you are getting 3 to 1 odds, because if you call, the pot becomes 750. So, you are putting in 250 rupees to win a pot of 750 rupees, thus:


750:250 :: 3:1


Why are pot odds important? Pot odds are compared to the probability of winning the hand. If your odds of winning the hand is higher than the pot odds, then your opponents bet is responded to by a call or a raise. Pot odds are divided into effective odds and implied odds. This article only deals with effective odds. While facing a raise, the odds that a player is getting to make a call with one card to come are effective odds. Implied odds, mostly used for drawing hands, are used to calculate the amount of money that you expect to win at the end of the hand if you hit one of your outs. This concept is beyond the scope of this article, and shall be explained in a future one.


Here’s an example of effective odds.


My cards - K ♠️ Q ♠️

Flop - 2 ♠️ 5 ♠️ J ♣️


As you can see, I have a flush draw on the flop, which means roughly, I have a 19.1 % probability of hitting the flush on the turn. A 19.1 % probability means I am getting 4.22 : 1 card odds. Keeping this in mind, let’s assume two situations.


Situation 1 


There are 300 rupees in the pot,and you are facing a pot raise, that is, a raise of 300 rupees. Now there are 600 rupees in the pot, and you have to make a decision. If you call, you are getting 3 to 1, that is, you are putting in 300 rupees to win a pot of 900 rupees.


Therefore,


Pot odds  → 3:1

Card odds → 4.22:1


The basic theory of poker says that when pot odds are lesser than card odds, a player must fold while facing a raise. On the other hand, when pot odds are greater than card odds, a player should either flat call or raise (raising builds implied odds). Therefore, in the aforementioned example, a player facing a 3:1 raise on the flop, with 4.22:1 odds, must fold. Let’s look at another situation with the same cards and same flop.


Situation 2


My cards - K ♠️ Q ♠️

Flop - 2 ♠️ 5 ♠️ J ♣️


Again, my card odds are 4.22:1, that is, a 19.1% chance of hitting a flush on the turn. But now consider the pot to have 1500 rupees. Your opponent bets 300 rupees. Now the pot size is 1800 rupees. You are facing a bet of 300 rupees, which means you will be putting in 300 rupees to win a pot of 2100 rupees. Therefore, your odds are 2100 : 300 or 7:1.


Therefore,


Pot odds → 7:1

Card odds → 4.22:1


Since pot odds > card odds, a player facing this bet should at least call as he is getting really good odds to win the pot. A player can also raise, in order to build implied odds, which are basically an extension of pot odds, telling us how much we expect to win after we hit our draw.

Effective odds are extremely important because they maximize our profits and minimize our losses. One should always remember that poker is a game of math, not pure chance. Knowing these simple concepts can go a long way in making an amateur’s poker career really successful. 

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