EV means “expected value”. In poker, this refers to the average amount of money a player can expect to win or lose by taking a certain action during a hand.

In this article, we will review the details of EV, including how to calculate it, how to use it effectively to boost your profits, and how it might be used in certain hypothetical examples that have been created.

HOW CAN I USE EV TO PROFIT IN POKER?

In poker, you want to (almost) always take the highest possible EV lines to maximise your profits.

The only time you may choose to not do so is when you’re trying to give yourself a higher EV over the long-term, perhaps by creating a certain image for yourself. (For example, when Gus Hansen called a 4-way all-in in PLO on Poker After Dark with his 8-6-3-2 single suit hand and 17.7% equity! In a later interview, he referred to this televised hand as part of his “advertising budget,” so he’ll get paid off in future instances against others who may have seen him play this.) In general, though, immediate, +EV decisions will always yield you the highest and most consistent profits over the long-term.

Often, when trying to determine the EV of an action, one will use a combination of a probable (yet uncertain) range analysis of their opponent(s) along with a logic-based prediction of the amount of time they expect their opponent(s) to call or fold.

Taking risks will be a big part of maximising your EV and profits (i.e. betting large as a bluff in ideal situations when or hero calling a hand after your opponent has bet or raised). Many players are afraid to take such risks, though, as they’re worried about the outcome of a specific hand.

Such players fail to realise that when they make certain plays, they shouldn’t be so results-oriented, but rather consider whether it’s a long-term, profitable play. As long as they are always getting their money in good and/or taking lines that yield a positive expected value versus specific opponents and their hand ranges in specific scenarios, then in the long-run, they will be winning players.

In summary, poker players should always be looking to maximise their long-term gains instead worry about short-term profits and taking risks (especially if these risks are +EV).

The EV Formula:

EV(Situation) = Percentage(X1)Amount(X2) + Percentage(Y1)Amount(Y2)

BASIC EV EXAMPLE #1

Imagine the following situation. Someone offers you the choice of two options:

  • Take $1,000 now.

  • Flip a coin, and if it lands on tails, you win $10,000. If it lands on heads, you win $0.

Many people I’ve asked this question, have said they’d happily take the $1,000. To most, this is a good chunk of change, adding that they’d “rather have something than nothing”. In other words, they’d rather take the guaranteed money and be averse to the risk of walking away empty-handed.

However, as a poker player, this is a classic situation where you should ALWAYS be happy and willing to flip the coin. But, why is this?

If you choose Option #1, 100% of the time, you will gain +$1,000.

If you choose Option #2, 50% of the time, you will win +$10,000 and the other 50% of the time, you will win $0.

We can write this as:

EV (coin flip) = P(tails)amt(tails) + P(heads)amt(heads)

EV (coin flip) = (50%)($10,000) + (50%)($0)

EV (coin flip) = $5,000 + $0

EV (coin flip) = $5,000

The EV of accepting Option #1 is +$1,000; and

The EV of accepting Option #2 is +$5,000.

Therefore, regardless of outcome, you should be ecstatic to flip the coin, as doing so will yield much more profit on average than choosing the guaranteed cash. In poker, and with EV, this is what we should be aiming for: maximum profits over the long-term. (Don’t be so fixated on short-term results.)

BASIC EV EXAMPLE #2:

One summer, Planet Hollywood’s poker room in Las Vegas had a high hand promotion, where if you made quads or better using both of your hole cards, you got a prize from the “Mystery Bag”.

This bag contained 6 tokens: 4 x $50, 1 x $100, and 1 x $200.

After selecting an initial monetary prize from the first bag, you then had an option of keeping your initial winnings or gambling by choosing one of two tokens from a second bag – one token which halved your winnings; another token which doubled your winnings.

QUESTION 1:

If you make quads or better, what is the EV of picking a token from the bag, assuming you’ll never use the second bag?

To answer this, all we must do is find an average.

EV = ($50 + $50 + $50 + $50 +$100 + $200) / 6

EV = $500 / 6

EV = $83.33

The EV you’ll win from the mystery bag every time you make quads or better is $83.33.

QUESTION 2:

Suppose you luck out and choose the $200 token. Should you choose to use the 2nd bag or keep the $200 as is? Is using the second bag +EV or –EV?

EV = p(choose ½ token)amt(lose $100) + p(choosing 2x token)amt(win $200)

EV = (0.5)(-$100) + (0.5)($200)

EV = -$50 + $100

EV = $50

Each time you choose to use the second bag after winning the $200 token, you should expect to have $50 more profit on average (giving you $250 altogether).

(You’ll notice that the $250 total is halfway between the two possible results of $100 profit or $400 profit. However not all EV calculations will involve this 50/50 formula, as we shall soon see.)

Give an amateur a choice of choosing from the 2nd bag, and they make not necessarily be willing to take the risk of potentially losing half. Instead, they might rather want to keep a hold of their guaranteed winnings.

 To a poker pro, though, they know that, in the long-run, they’ll be profiting on average each time they choose to use the 2nd bag. (Remember, immediate results do not matter to poker pros.) Subsequently, serious players should always confidently choose to use the 2nd bag as this will yield them a higher EV (or average result).

Poker player sitting at his computer thinking about complex math equation

SO, WITH EV - WHAT DOES THIS ALL MEAN?

If every decision you make in a poker hand maximises your expected value, you will begin to start crushing the game with incredible results!

Subsequently, it’s important to set aside some time “in the lab” so that you can study examples and concepts regarding EV. (Yes, it will be time-consuming and perhaps difficult at the start, but as with all things, it will get easier with practice. Furthermore, your results at the table will skyrocket if you take the time away from the table to study and improve on this stuff.)

The study that you may choose to do regarding EV can either be like the practical, hand-written examples we’ll go through later in this article, or with the assistance of solvers that you can purchase online (such as PokerSnowie and PioSOLVER).

These programs greatly help simplify the process of seeing what different EV’s one can expect by taking different lines (fold/check/bet/raise), and by considering different bet or raise sizings, too!

 

EQUITY AND EXPECTED VALUE (EV)

Before we get into the examples below, it’s important to clarify the difference between equity and expected value, as sometimes these two terms are incorrectly used interchangeably.

Equity (unlike EV) refers to the amount of money in the pot that should belong to you based on the percentage of how likely your hand is to win at showdown versus your opponents.

For example. let’s say that you get QQ all-in preflop vs. AKs. In this situation, the QQ is a small favourite, and we would say that it has a 54% equity advantage. (Alternatively, we could say that in a $400 pot, the equity that QQ would have versus AKs has is (54%)($400) = $216. )

If we were calculating the EV of this specific situation, we would use the knowledge we have about the equity (percentages) to assist us in finding the outcome.

EV = pWinning(54%)amtWon($200) + pLosing(46%)potLost($200)

EV = $108 - $92

EV = $16

With effective stacks of $200, the EV of getting QQ all-in preflop against AKs is +$16, meaning you’ll profit an average of $16 whenever this situation occurs.

EXAMPLE #1: BET SIZING

EV is not only about profiting and losing. It’s also about maximising expectations on the amount of money you can win (or save from losing when you’re caught bluffing).

Bet sizing, therefore, is a very important part of applying the concept of EV into your poker game and winning the most money possible.

In this example, we’ll imagine we have the nuts on the river. With strong holdings, some opponents will choose to bet small to do everything to induce a crying call from their opponent. In many situations, making a large bet with such a strong hand will be vastly superior, even if you think your opponent will call only a slightly lower percentage of the time (as we’ll soon see)!

Imagine there’s $200 in the pot on the river and you have A-J on A-3-J-J-9. Your opponent has checked and there’s $400 remaining in your stack. What is the EV of using a bet of $100 vs. an all-in bet of $400? In which situations should we use one sizing over the other?

In this bet sizing instance, it all comes down to your perception of how often your opponent will call each size of bet. For example, let’s say that your opponent will call a $100 bet 50% of the time, and a $400 bet only 25% of the time.

Which bet carries with it the higher expected value?

$50 River Bet

EV = pOppCalls(50%)amtPotPlusBet($300) + pOppFolds(50%)amtPot($200)

EV = ($150) + ($100)

EV = $250

$250 River Bet

EV = POppCalls(25%)amtPotPlusBet($600) + pOppFolds(75%)amtPot($200)

EV = ($150) + ($150)

EV = $300

Here, we can see that even though our opponent will call a bigger bet a smaller percentage of the time (25% vs. 50%), the EV of overbetting is greater than that of betting $100, providing our assumed percentages of how frequently our opponent will call are correct.

EXAMPLE #2: VALUE  BET VS BLUFF CATCH

Much of the concept of EV comes down to assumptions based on percentages and likelihoods of how your opponent will react in certain situations.

Let’s say that the board is Ac-4c-8s-9d-2s. We have AKo. We have a couple of notes on our opponent:

  • He will value bet weaker hands than he will call.
  • He will often fold his weakest top pairs to triple barrels.
  • He will only call on the flop and turn with his draws when in position.
  • He will bluff his missed draws when checked to but will rarely raise the river as a bluff.

In this situation, we have raised pre, cbet the flop, and cbet the turn with top pair, top kicker. (Our opponent just called both times.) Now we must decide whether it’s better to bet or check, regarding which option yields a higher EV.

CHECKING VS. BETTING:

By checking, we can likely get our opponent to bet with a much wider range of hands than he would call a river bet:

  • His missed draws
  • Some weaker top pairs
  • Best top pairs (that we still beat)
  • Hands that beat us

Compare this to the list of hands we would get called by if we bet:

  • Best top pairs (that we still beat)
  • Hands that beat us

Therefore, you can see that because we know our villain over-bluffs his missed draws (and all draws except the unlikely 53s bricked on the river), we can gain a higher frequency of bets from our opponent than calls if we bet.

Now let’s try to make some further assumptions and calculate the EV of checking vs. betting:

  • Villain will bet 80% when we check to him
  • Villain will call only 20% of the time if we bet.
  • In both cases, we think we will win 90% of the time
  • The pot is $150 and remaining effective stacks are $100.
  • Any further bet would be for $100.

CHECKING:

EV(checking) = P(Villain Bets)P(WeWin)amt(Won) + P(Villain

Bets)P(WeLose)amt(Lost) + P(VillainChecks back)amt(WonFromPot)

EV = (80%)(90%)($250) + (80%)(10%)(-$100) + (20%)($150)

EV = (0.72)($250) + (0.08)(-$100) + (0.2)($150)

EV = $180 - $8 + $30

EV = +$202

BETTING:

EV(betting) = P(Villain Calls)P(WeWin)amt(Won) + P(Villain

Calls)P(WeLose)amt(Lost) + P(Villain Fols)amt(won)

EV = (20%)(90%)($250) + (20%)(10%)(-$100) + (80%)($150)

EV = (0.2)(0.9)($250) + (0.2)(0.1)(-$100) + (0.8)($150)

EV = (0.18)($250) + (0.02)(-$100) + (0.8)($150)

EV = $45 - $2 + $120

EV = +$163

Based on our assumptions, we can see that checking and inducing our opponent to bluff a high percentage of the time will yield a higher EV than betting in this particular instance.

NOTE: To arrive at more clear percentages with our assumptions, we can do a range analysis for our opponent and determine exactly which and how many combos we think he will (1) call if we bet or (2) bet himself if we check to him, relative to all the possible combos in his entire range.

The next example will use some combination and range analysis to determine more accurate EV’s and results.

EXAMPLE #3 (ADVANCED): BLUFFING VS RIVER SHOWDOWN VALUE

Assume we have AcQh on Kd-Jh-7d-2c-Qc. We 3bet preflop in position and our opponent called. He proceeded to check-call both the flop and the turn. There’s $160 in the pot and we have $120 left in our stack.

We must decide whether first whether it’s profitable to bet this river as a bluff, and then determine which action carries the highest EV: bluffing the river or checking-back our rivered 2nd pair with showdown value.

To do this, we must assign a range of hands to our opponent and determine how many of those combinations we beat already. Then, we must determine which of these hands we think he will fold or call to determine the EV of checking back or betting as a bluff.

A great deal of this example will come from our assumptions of our opponent’s range and tendencies, many of which will originate from his preflop stats or any previous notes we’ve taken on him.

Here’s our assumptions of his range:

 

WHAT VILLAIN HAS

COMBINATIONS

DO WE BEAT IT?

DO WE THINK HE WILL FOLD BETTER HANDS IF WE BET?

AKo                            

7 combos

No

Yes

AKs

2 combos

No

Yes

AJs

2 combos

Yes

n/a

ATs (hearts only)

1 combos

No

No

KQs

2 combos

No

No

KJs

2 combos

No

No

QQ

1 combos

No

No

QJs (hearts only)

1 combos

No

Yes

JJ

3 combos

No

No

JTs (hearts only)

1 combos

Yes

n/a

T9s

4 combos

Yes

n/a

TOTALS:

26 combos

7 combos

10/19 combos

Note some of the assumptions:

  • We do not think villain would have 4bet us preflop with AK. (Having these combos for him to be able to fold out is quite important in this instance.)
  • If we bet, we think villain will fold all the hands we beat, in addition to 10 other combos (of the 19 remaining).
  • After 3betting and triple-barrelling, our range is usually very strong here (top pair, two pair, sets, straights) and we do not have many bluffs (maybe some suited connectors like T9s, if we play them as 3bets). Because of this, we think villain will fold his AK hands to an additional bet, in addition to his weakest two pair combination of QJs.

To find out how many combos villain needs to fold out to determine if this bluff will profitable, we must take…

  • …our remaining stack size…
  • …the pot size…
  • …how many winning combinations villain has…
  • …how many of these he will fold out…

…and put them all together.

By betting $120 into $160 (75% of the pot), we see that we need to win 43% of the time for this to be profitable. (See chart in next section.)

43% x 19 combinations that beat us means he must fold 8 of those combos for this bet of ours to be profitable. Because we determined that he will likely be folding 10 combinations of hands, we can bet here. HOWEVER, just because betting here as a bluff will be profitable doesn’t necessarily mean that it will be the most profitable option.

Our hand also has showdown value, now that we improved to 2nd pair! If we check here, we automatically win against 7 of his 26 total combinations.

This scenario means that the equity of checking is:

EV (checking) = (7/26) x size of pot

EV (checking) = 0.269 x $160

EV (checking) = +$43.08

Let’s compare this now to the EV of betting:

EV (betting) = Pwin(10/19)amtWon($160) + Plost(9/19)amtLost(-$120)

EV (betting) = Pwin(52.6%)amtWon($160) + Plost(47.4%)amtLost(-$120)

EV (betting) = $84.16 + (-56.88)

EV (betting) = +$27.28

Even though we correctly determined that checking and betting as a bluff will both be profitable here, using the concept of EV, we were able to determine that checking has a higher expected value in this situation.

SUPPLEMENTAL CHARTS:

Here are two charts to assist you in understanding percentages of how often you must win the pot when you bet or when calling a bet.

These can be used to help you out in certain EV calculations:

VILLAIN’S BET

WE MUST WIN

 2x pot

40%

1.5x pot

37.5%

Pot-size

33%

¾ pot

30%

⅔ pot

28%

½ pot

25%

⅓ pot

20%

¼ pot

16%

 

OUR BET

WE MUST WIN (VILLAIN FOLDS)

 2x pot

66%

1.5x pot

60%

Pot-size

50%

¾ pot

43%

⅔ pot

40%

½ pot

33%

⅓ pot

25%

¼ pot

20%

UNDERSTANDING VARIANCE

Always remember that EV is the average amount you will profit or loss from taking a certain action. Often, the wins or losses will be much greater in a hand than the EV dictates. However, with enough volume, your expected profits and losses should come very close to matching your actual profits or losses.

If you make a +EV play and end up losing the pot, don’t get discouraged, and don’t tilt. There’s always another hand, and – as previously mentioned – the sum of your results will ultimately be made up of the averages of all the hands you’re involved in and the decisions you make, and not just the big ones.

SUMMERY

Understanding Expected Value (EV) can not only show you which plays will be profitable, but also exactly how much more profitable certain actions and different bet or raise sizings will be - relative to each other.

Now that you have been armed with the knowledge of the EV formula, start to use it in some examples in your own. Work out what is the highest +EV play to make in certain situations, and perhaps which leaks you might need to start plugging in your game to boost your profits.

Remember, that computer software such as PokerSnowie and PioSOLVER can also assist you with EV calculations and correctly determining the most profitable lines to take in poker.

 

About the Author
By
Matthew Cluff is a poker player who specialises in 6-Max No Limit Hold’em games. He also periodically provides online poker content for various sites.
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