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Calculating Expected Value

math, poker and EV.It is important to know how to calculate expected value in common poker situations in order to make better decisions in the future.  The formula for calculating expected value in general, which is also used for calculating expected value in poker is as follows:

1. Multiply each possible outcome by the probability of that outcome occuring

2. Add each outcome

The result is expected value.

Example EV Calculation: Semi Bluffing

Player 1 holds A♠Q♠ on a board of K♠8♠2♥.  With $100 in the pot and $400 effective stacks, Player 2 bets $100.  Player 1 shoves all-in as a semi-bluff.  Lets say Player 2 only calls with AK, KQ, KJ and sets.  Player 1's equity against this range is approximately 40%.  Lets also say that Player 2 will fold 50% of the time.  Thus:

    • 50% of the time Player 1 wins $200 (the $100 in the pot plus Player 2's $100 bet).
    • Of the other 50% of the time, Player 1 will lose $400 (his stack) 60% of the time and win $500 (his stack plus the pot) 40% of the time.

The equity calculation for Player 1's shove would then look like this:

EV=(.5)(200)+.5[(.6)(-400)+(.4)(500)]= 80

This means Player 1 can expect to profit $80 each time he makes this bet.

Example EV Calculation: Continuation Betting with Air

In a $2/5 game with $500 effective stacks, Player 1 has raised to $20 from late position with a steal hand and Player 2 has called from the big blind.  The pot is $40 after the rake.  The flop comes A♠8♥2♣, Player 2 checks and Player 1 makes a continuation bet of $30.  

If Player 2 folds 65% of the time, what is the EV of Player 1's c-bet?  (Assuming that Player 1 must give up if his c-bet is called and that he will never win the pot.)  

In this scenario, Player 1 will win $40 65% of the time and lose $30 35% of the time.  The EV calculation is as follows:

EV=(40)(.65)+(-30)(.35)= 15.5

This means that Player 1 can expect to profit $15.50 each time he makes this continuation bet.  While a simple calculation, this is an important example because continuation betting in heads up pots is one of the most common situations in No Limit Texas Hold'em.

Example EV Calculation: Value Betting

It is a $2/5 game with $1000 effective stacks.  On the river the pot is $200 and the board reads K♥6♠4♠6♣9♥.  It is heads up and Player 1 checks to Player 2 who is pondering a value bet with K♦Q♦.  He could check behind and win the pot 90% of the time.  This means he will win $200 90% of the time and will not win or lose anything the other 10% of the time (remember, the money he has put in the pot is no longer his).  Player 2's EV of checking is therefore simply (.9)(200)= $180.  

However, if he bets $100 he will be called by Player 1 40% of the time.  10% of the time is when Player 1 has a better hand, but 30% of the time is when Player 1 has a worse hand that is worth calling a bet.  It is assumed Player 1 will never fold a better hand and will never check/raise as a bluff.   So there are 3 possible final outcomes when Player 2 bets:

  • 60% of the time Player 1 folds and Player 2 wins $200-Outcome 1
  • 40% of the time Player 1 calls.  Of this 40%;
  • 75% of the time Player 1 loses and Player 2 wins an additional $100 ($300 total)-Outcome 2
  • 25% of the time Player 1 wins and Player 2 loses $100-Outcome 3

The Equation becomes:

EV=(.6)(200)+.4[(.75)(300)+(.25)(-100)]= 200

Therefore, Player 2's EV of betting is $200.  Since this is higher than the $180 EV of checking, Player 2 should bet.  Even though betting makes it possible to lose additional money, it is still better in the long run.  This is true because when called Player 2 has the best hand greater than 50% of the time.  However, if he had the best hand less than 50% of the time when called, checking would be better.

Expected Value in Pot Limit Omaha

Omaha EV - Not always fun!

Calculating expected value in Pot Limit Omaha (PLO) can be very complex.  Many of the most common scenarios involve semi-bluffing, the calculations for which are identical to the semi-bluffing section above except the hand equities are different, which is illustrated in the example below.

 Battle it out at PLO with other US players at True Poker.

Flop All-ins

Flop all-ins are much more common in PLO than holdem.  Most of these involve a big made hand, often a set or a straight, getting all in against a big draw.  For example, with a $100 pot on a flop of K♠Q♥2♠ Player 1 bets $100 holding A♠J♠T♦8♦ for a 13-card straight draw and nut flush draw.  Player 2 moves all-in for $400 holding K♦K♥8♥7♦ for top set.  What is Player 1's EV of calling?

Despite only having ace high right now, Player 1 actually has 49% equity against Player 2's set because of all his draws.  He has to call $300 more to win the $600 in the pot (400 from Player 2, his original $100 bet and the original $100 pot).  Therefore he will win $600 49% of the time and lose $300 51% of the time.  His EV of calling is as follows:

EV=(.49)(600)+(.51)(-300)= 141

Thus Player 1 can expect to win $141 each time he makes this call.  He will lose just over half the time though, which makes this bet very high variance.

Cracking AcesThe EV of Cracking AA

A fairly common scenario that is unique to PLO is when a player calls pre-flop, usually when facing a 3-bet or 4-bet, when he knows the other player has aces.  Unlike in holdem, this is often correct in PLO because the hand equities run so close together.

It is an online 6-max $1/2 PLO game with $200 effective stacks.  Player 1 raises from MP to $7.  Player 2 3-bets from the button to $24 holding a gappy double-suited rundown such as J♦9♠7♦6♠.  Player 1 now 4-bets to $75.  The pot is now $102 and it is $51 for Player 2 to call.  If he makes the call the pot will be $153 and there will be $125 left to play.  Based on previous play, Player 2 knows that Player 1 must have aces.  He also knows that Player 1 will bet his remaining $125 on the flop 100% of the time.

On the flop, Player 2 will need to call 125 to win 278, so he must have at least 125/(278+125)=31% equity to call.  He will flop at least this much equity roughly 60% of the time.  Because Player 2's equity is sometimes higher than the required 31%, his average equity when he calls is approximately 55%.  We now have all the information to calculate the EV of Player 2 calling the 4-bet.

  • It costs $51 to call the 4-bet.  40% of the time, he calls the 4-bet and folds the flop for a net loss of $51.
  • 60% of the time, he calls the 4-bet and calls the flop shove.
  • Of this 60%, 45% of the time he loses an additional $125 for a total loss of $176.
  • 55% of the time he wins the $102 pot he is facing when deciding whether or not to call the 4-bet, plus an additional $125 for a total of $227.

The calculation is as follows:


Therefore, the EV of Player 2 calling the 4-bet is $20.76, so he should make the call.  

Clearly calculating EV in poker can range from very simple to extremely complex.  Knowing how to do and understand these calculations will make it easier to consistently make positive EV decisions and win money in the long run.

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