I am wondering how to mathematically calculate equities in equity based poker variants( holdem, omaha, Stud ect).
In general, equity is your expected value when you would run the same hand infinite times. In probability theory, we take the calculated chances as truth in a scenario where the event is repeated infinitely.
In poker, your equity or expected value can be calculated by taking your chance of winning and multiplying that with the value of the pot. We have in this case two scenario's:
You currently have the best hand
When you already have the best hand, you can only lose if your opponent(s) hit their out, which means they improve to having the best hand. Therefore, calculating equity is cumbersome since you do not know the hand of the opponent. You can however 'read' his hand based on the board, betting tells, past behaviors, etc. When you have a solid read, say he is drawing to a flush on the flop, you know that he has a chance of 0.35 to improving and therefore you have a winning chance of 0.75 ( with for example a pair of aces.
This is, however, a very loose approximation since he may also hit two outers like two pairs or may also draw to a straight etc. In general: read the hand, approximate his outs and calculate his chance of improving. Subtract this probability from 1 and you have your probability of winning.
This probability of winning can then be multiplied with the pot, say for example $100,- and you will get your equity.
You currently do not have the best hand
When you do not have the best hand, you can just calculate your own chances of hitting your outs. Of course, if you are drawing to a flush but your opponent is drawing to a full house you may have to take this into account, but that is a very hard read to do.
In another answer I explain how you can calculate your probability of winning on the flop or turn. When you have this probability, you can multiply it with the pot to get your equity.
--EDIT-- The answer where the hand of the opponent is known
To calculate the equity when you know the hand of the opponent, I am going to use the following events:
- E1: You hit on the turn
- E2: You hit on the river
- E3: Opponent hits on the turn
- E4: Opponent hits on the river
When the opponent hits, it beats your hit. Therefore, if E1 and E3 happen, you lose. If E1, E2 and E3 happen, you win, since you hit something on the river which beats the hit by your opponent.
Winning chance is in this case combining the events. The ‘-’symbol is used as a negative and hence interpreted as -event does not happen-. We can then calculate the winning change by combining the events the following way:
E1 * -E3 * -E4 + E2 * -E4
For example, you have K♦Q♦ and your opponent has A♠A♥ with a board A♦5♦J♠
You have 7 outs to hitting a flush (not the J♦ and not counting 10♦), three to a straight (not counting 10♦) and one to royal flush, hence:
E1 = 11/47 = 0.23
Now we can calculate the probability that you hit the flush or straight on the turn. Note that I count 10 outs since 10♦ would give you royal flush.
E2.1 = 10/46 * (1 - -E1) = 0.17
Now we have to subtract the probability that the full house fell which makes this flush or straight obsolete:
E2.2 = 0.17 * 0.85 = 0.14 (see below for how I get 0.85)
However, we can add the probability that the 10♦ fell, giving you royal flush:
E2 = 0.14 + 1/47 + 1/46 * (1-1/47) = 0.14 + 0.02 + 0.02 = 0.18
He has 7 outs to improving (1 ace, 3 fives and 3 jacks). Therefore:
–E3 = 1-7/47 = 0.85.
E4 is the event that E3 did not happen and that opponent improves. Therefore:
E4.1 = 7/46 * (1-7/47) = 0.13
However, also the 10♦ should not have fallen on the turn and river:
E4 = 0.13 * 0.98 * 0.98 = 0.12
Which makes -E4 of course:
-E4 = 1-0.12 = 0.88
Filling in the numbers:
- Hitting the turn: 0.23 * 0.85 * 0.88 = 0.17
- Hitting the river: 0.18 * 0.88 = 0.16
Which gives you a winning chance of 0.33, which is quite lower than the ‘normal’ out-hitting approximation of 0.48 (12 outs * 4)
You equity will than, of course, be 0.33 * pot-size.
You can check the above scenario at an odd calculator, which gives 32,93% which is rounded of course the same as 33%