Any action you make in poker game should be focused on making a profit. Your efforts are worthwhile if they lead to the maximum profit possible or have the best average profit value. The average value of a possible profit is also named Expectation Value (EV).

Sometimes there are situations when you have several possible ways to play the hand. The profit of each variant = ev1, ev2,...,evN, the possibility of each variant = p1, p2,..., pN accordingly. Then the expectation value of the profit in the given situation is calculated by the formula:

EV = ev1p1 + ev2S2 + ... + evNSN ()

The following is a very complicated example, but it has great practical value and touches on all the basic moments and rules of calculation. Let us assume that you have AK and raised on the preflop. The contender re-raised. Both stacks are 100BB. What is your best move? Should you call his re-raise, fold or re-raise (4bet)?

To answer these questions you need to calculate the EV of the call and the re-raise. The EV of the fold, obviously, is zero.

It is necessary to note that the EV of an action is the difference of our stack between the end and the beginning of an action, but not at the beginning of a hand. You can’t get the money you have just invested in the pot.

When you call:

Let us set a task first:

The opponent made 3bet of your raise (raise=5??, 3bet=15??). You need to determine the possibility of your opponent having certain hands.

Let us divide the range of 3bet into two groups

The first group: he will play 3bet every time; these are QQ+ and ??.

The second group: he will play 3bet less than 100% of the time.

JJ: 70%. It means that he will re-raise with JJ 70% of the time, and will not make any further play.

AQ, ??: 30% (no more for small limits)

AJ, QK: 20%

1)?? and ?? have three combinations each, QQ-6, ??- 9, total 18

2)JJ - 6

3)AQ - 12, ?? - 6, total 18

4)AJ, KQ - 12 each, total 24

The total number of hands the opponent may re-raise with

13+60.7+180.3+24*0.2 = 32,4

The possibility the opponent has ??:

?(??) = 3/32.4 = 0.09.

The possibility of other hands:

?(??) = ?(??) = 0.09.

?(QQ) = 6/32.4 = 0.19

P(AK) = 9/32.4 = 0.28

P(JJ) = 0.7*6/32.4 = 0.13

P(AQ) = 12*0.3/32.4 = 0.11

P(TT) = 6*0.3/32.4 = 0.06

P(AJ) = P(KQ) = 12*0.2/32.4 = 0.07

Let’s calculate the EV of various possibilities:

AA:

If you have K and/or A on the flop you should play with your entire stack.

The flop with a king and/or an ace may happen 23% of the time.

You may calculate this in the following way: one ace and three kings are left in the pack, four cards. To calculate the required probability, let’s calculate the probability of the opposite situation - there will be neither an ace nor a king on flop.

The probability that the first card from the flop is neither an ace nor a king is ?1 = 44/48, the second P2 = 43/47, the third P3 =42/46. These probabilities are practically equal, so let us use the number 44/48.

The required probability of the opposite event

P = P1P2P3 ~= P1^3

The probability of the initial event is

1-? = 1-(44/48)^3 = 0.23

(Further similar calculations will be omitted)

The rest of the time (77%) you will lose 10BB, because you will lay your cards down on a cont bet.

EV (??) = -100.77 + (- 950.23) = -29,5 BB

??:

If there is an ace on the flop without a king you will win the pot + cont bet 2/3 of the pot.

The probability of this kind of flop is 16%.

Thus, at least one card from the flop should be an ace and the other two cards shouldn’t be a king. You should find the product of these probabilities. The probability of the first event is calculated like in the previous example.

P1 = 1 - (45/48)^3 = 0.18

The probability of the second event:

P2 = (45/47)^2 = 0.92

The required probability:

? = ?1*?2 = 0.16

(Further similar calculations will be omitted)

On the flop with a king you lose your stack. The probability of the flop with a king is 6%.

The other 78% of the time you should lay your cards down against a cont bet.

EV (??) = (20 + 302/3)0.16 - 950.06 - 100.78 = -7.1 BB

QQ:

You will win the pot + cont bet on a flop which brings you an ace or a king. The Probability of this flop is 33%. Otherwise, you should fold. A flop with a queen and an ace and/or a king is highly unlikely and you shouldn’t count on it.

EV (QQ) = 0.33*(20+302/3) - 0.6730 = -6.9 BB

AK:

You lose nothing on a flop with an ace or a king. The Probability of this happening on the flop is 23%. On other flops you should fold against a cont bet.

EV (AK) = - 0.77*10 = -7.7 BB

JJ, TT:

EV (JJ) = ev(TT) = ev(QQ) = -6.9 BB

AQ and AJ:

Calculation:

EV (AQ) = EV (AJ)

Don’t take into consideration the possibility of two pairs from the both sides.

You will win the stack on a flop which draws an ace. The probability of this occurring on the flop is 12%.

You will win the pot + cont bet on a flop with a king, but no ace.

The probability of this type of flop is 16%.

The other 72% of the time you should fold against a cont bet.

EV (AQ) = EV (AJ) = 0.12105 + 0.16(20+302/3) - 0.7210 = 11.8 BB

KQ:

You will win the stack on a flop with a king, but no ace.

The probability of this flop is 11%.

You will win the pot + cont bet on a flop which brings you an ace.

The probability of this flop is 18%.

In all other situations you should fold.

EV (KQ) = 1050.11 + (20+302/3)0.18 - 0.7110 = 11.65 BB

Total EV:

EV = EV (AA)*P(AA) + EV (KK)*P(KK) + EV (AK)*P(AK) + EV (QQ)*P(QQ) + EV (JJ)*P(JJ) + EV (TT)*P(TT) + EV (AQ)*P(AQ) + EV (AJ)*P(AJ) + EV (KQ)*P(KQ) = - 5.2 BB

As we can see, under the given conditions and assumptions the 3bet call with ?? won’t bring a profit.

Now let us consider the possibility of a subsequent increase, with the same entry conditions and assumptions.

After the 4bet, you play poker on the stack or the contender folds.

AA:

The probability of winning against ?? is 8.4% (AK suited and off-suite are taken into consideration).

This number can be reached by using a calculator.

AK of the same suit and off-suit must be taken into consideration. You may use the calculator on our site for this purpose. Having considered the probabilities of a prize with ??s and AKo, the final probability can be calculated with the formula P = 1/4* P(AKs) + 3/4* P(AKo)

EV (AA) = -950.916 + 1050.084 = -78.2 BB

P(AA) = 0.09

KK:

The probability of winning against KK is 30.3%

EV (??) = -950.67 + 1050.303 = -31.8 BB

P(KK) = 0.09

?K:

Here it is necessary to assume that the probability of AK folding after 4bet is approximately 50%.

EV (AK) = 0.5*20 = 10 BB

P(AK) = 0.28

QQ:

The probability of QQ folding after 4bet is approximately 50%.

The probability of winning against QQ is 44%

EV (QQ) = 0.2520 + 0.75(-950.56 + 1050.44) = -0.25 BB

P(QQ) = 0.18

JJ:

The probability of JJ folding after 4bet is approximately 70%.

The probability of winning against JJ is 43%

EV (JJ) = 0.720 + 0.3(-950.57 + 1050.43) = 11.3 BB

P(JJ) = 0.13

TT:

The probability of TT folding after 4bet is approximately 80%.

The probability of winning against TT is 44%

EV (TT) = 0.820 + 0.2(-950.56 + 1050.44) = 14.6 BB

P(TT) = 0.06

AQ:

The probability of AQ folding after 4bet is approximately 80%.

The probability to win against AQ is 70.2%, to lose - 23.7%.

EV (AQ) = 0.820 + 0.2(-950.237 + 1050.702) = 26.2 BB

P(AQ) = 0.11

AJ:

The probability of AJ folding after 4bet is approximately 85%.

The probability of winning against AJ is 69.4%, to lose - 24.3%.

EV (AJ) = 0.8520 + 0.15(-950.243 + 1050.694) = 24.5 BB

P(AJ) = 0.07

KQ:

The probability of KQ folding after 4bet is approximately 90%.

The probability of winning against KQ is 73.7%, to lose - 24.7%.

EV (KQ) = 0.920 + 0.1(-950.247 + 1050.737) = 23.4BB

P(KQ) = 0.07

Total EV:

EV = EV (AA)*P(AA) + EV (KK)*P(KK) + EV (AK)*P(AK) + EV (QQ)*P(QQ) + EV (JJ)*P(JJ) + EV (TT)*P(TT) + EV (AQ)*P(AQ) + EV (AJ)*P(AJ) + EV (KQ)*P(KQ) = 1.7 BB

The total EV is quite low. Our calculations were based on your opponent playing tight. We excluded all pocket pairs lower than ?? and weaker aces, but that doesn’t mean you won’t come across them. Therefore it is possible to assume, that a draw of that kind is ideal, though not very profitable. In this way you still maintain a positive image. It is possible to come to the conclusion that playing 4bet with AK when the stacks are 100BB against an opponent you are not familiar with is more profitable than calling 3bet raise or folding.

The given formulas, at first glance, are a little difficult, but if you have a desire, it is simple to understand them. If it is necessary to calculate something you can try an analogy. This information is not necessary, but always useful in understanding the poker game. The term EV can be seen in articles and on forums, but, unfortunately, many authors often don’t understand its value.

Good luck!