There are 1,326 possible combinations of cards from a standard deck but there are only 169 non-equivalent starting hands in poker. This number is made up of 13 pocket pairs, 78 suited hands and 78. What is the highest hand and hands order in poker? You can see the hands order below starting with the highest ending with the lowest:. Royal Flush: 10, Jack, Queen, King, Ace all in the same suit. Straight Flush: Five cards in a row, all in the same suit. Four of a Kind: The same card in each of the four suits. Full House: A.

Expected Value (EV) is the average return on each dollar invested into a pot. If a player can expect, given probability to make more money than he or she bets, the action is said to have a positive expectation (+EV). Conversely if a bet or a call will, according to probability, likely result in less money being returned the action is said to be negative (-EV).

An example may assist in the understanding of this concept. In Texas Holdem it is quite common for someone to flop 4 to a flush. The person should only draw to that flush if to do so would be +EV. In order to calculate the EV it is necessary to compare the size of the bet with the size of the pot. A flopped flush draw will come in approximately 1 in 3 times by the river, thus in order for a call to be +EV the final pot must be larger than 3 times the call. This is a complicated issue so it may be useful to elaborate with a specific example.

Say you are playing 5-10 limit poker on the button, there are 3 limpers to you and you call with A4 diamonds. Both Blinds call so there is $30 in the pot. You flop the nut flush draw. The player in the small Blind bets $5 and there are four callers. Should you call, raise or fold?

Well, there is now $55 in the pot and it will cost you $5 to call so the pot is giving you 11:1 odds (i.e. you must pay $5 to win $55). We already know that the flush draw will get there 1 time in every 3 (2 to 1) so making the call is +EV. However, calling is not necessarily the best play in this situation. If you raise and the other 5 people in the pot decide to call your raise then you will be adding $5 to the pot whilst they will collectively be adding $25. This ratio is 5:1 but the chance of making your flush is only 2 losses to 1 win, so on average in the long run you are making money from every extra bet from all 6 of you that goes into the pot. Notice that even though you have only a 33% chance to win the pot, the correct thing to do is actually to bet, despite knowing that you will probably not win that particular single hand: you will win about 1 in 3 such hands in the long run, minus the few percent of the time when someone beats your flush with a full house or quads or straight flush.

Notice that all that has been done so far is compare the current pot with the bet size needed to call to calculate EV. However it is important to also compare the expected pot size by the end of the hand with the current bet. For example say you are playing no limit holdem and have a gutshot straight draw (giving you 4 outs to complete - approximately 1:12 against). If the pot is $30 and you are faced with a $10 bet the pot is not giving you the correct odds to call (it would need to be $120 total, plus your $10 call). However you also need to take into account the amount of money you may be able to extract from your opponents if you make your hand. If you expect your opponent to call a $100 bet if you make your hand, then the pot is really offering you 13:1 odds (the $30 pot at the time plus the $100 added on later streets) Therefore in this situation the +EV play would be to call. Thus when making decisions about whether to call a bet it is crucial to take into account both the stack sizes of yourself and your opponents and how willing they are likely to be to call big bets if you make your hand.

To make it easier to understand why this move is correct even though it usually loses, suppose you have a six-sided die. If you correctly guess what side it lands on, you will win $50. If you are wrong, you lose $5. You will be wrong five times out of six, but you stand to gain a lot over the long run! This is because the probability of guessing correctly is 1/6, sometimes expressed as odds, '5:1 against' (five losing possibilities, one winning possibility). However, the payoff odds are 50:5 ($50 won for a $5 bet), which can be reduced to 10:1, and 10:1 is twice as large as 5:1. The payoff odds are called pot odds in a poker game. Comparing the odds of winning to the pot odds is how you can estimate your expected value.

Ideally, you want to avoid all situations where you have a negative expectation. Even slightly negative situations can pile up and bleed away your bankroll. Casinos worldwide make MILLIONS of dollars lost by players against a 0.6% craps dice game house edge: even 0.1% is enough of an edge to wipe out all the billions of dollars of the richest man on earth, over time in the long run, which is why Bill Gates wisely bets only $5/hand for fun at blackjack!

Calculating expected value[edit]

You cannot always get a good idea of the chances of winning your hand & calculate the pot odds: at least, not without knowing what your opponents have, and they're not going to tell you! However, you will often have a draw which, if you hit, you will very likely win the pot. The exact arithmetic involved varies from game to game. In Texas hold'em and Omaha, once you see the flop, the percent chance of making your hand within one card is generally your number of outs (cards that will make your hand) multiplied by two, and the odds of making your hand within two cards is your number of outs multiplied by four. For example, if you have four hearts and you need one more for a flush, you have nine outs, because there are thirteen hearts in the deck, and subtracting the four hearts you already have gives nine. 9 × 2 is 18, so you have about an 18% chance of making the hand in the next card, and 9 × 4 is 36, so you have about a 36% chance of making it in two cards.

To make this easy, you want to turn this percent chance into odds, like 5:1 against. Fortunately, they are easy enough to memorize:

The odds in bold are the most important to commit to memory; the others can be easily estimated.

Now, take the x in the x:1 figure and multiply it by the bet size. For example, if the odds of making your hand are roughly 4:1, and the next bet costs $5, multiply 5 × 4 = 20. That means you want there to be at least $20 in the pot (be sure to include bets that have not been added to the pot proper yet!), preferably a bit more just in case unless you're certain to win if you hit your draw. If there is not at least $20 in the pot you will lay down your hand, unless you can check instead. If the table is really loose, and a lot of players are in the hand and are likely to stay in, and the pot will get really big, you may even want to raise. Normally, however, checking or calling is the correct move.

Notice we did not calculate the exact expected value. This is not necessary or indeed practical for most people. If it is negative, you get out, and if it is positive, you call. If you're a favorite to win the pot, you raise. However, as has been shown you can usually figure out if the value is only barely positive, for instance, the size of the pot is a dollar more than the odds of making your hand (and this dollar is small in proportion to the pot size). When faced with this situation, you might want to lay down your hand sometimes: you may be losing just a little money in the long run, but you keep your bankroll from taking big swings. But if you don't mind taking a gamble, by all means go for it!

Expected value (EV) is a way of calculating how much we stand to make in a particular situation – it can be applied to poker or in lots of areas in real life.

Table Of Contents

Expected Value Definition

The definition of expected value is the average returns we would expect from taking a particular action (.i.e betting/raising/calling).

The expected value will be based on our current pot equity and pot odds (i.e. opponents bet size) when we face a bet and is based on our equity, our betsize and our oppoents fold frequency when we bet or raise.

If we are the player making the bet, the expected value will be based on our pot equity, bet size and fold equity.

Expected Value: Real Life Problem

To take a real world example – we park our car in a city and unfortunately it costs is $5 an hour. We know we will be at least 1 hour.

We estimate that the probability that we will be caught without a ticket is 10%. The fine for not having a parking ticket is $60.

Should we buy the ticket beforehand?

Or take the chance that we will not be caught?

To solve this problem we can use expected value.

We can find the expected value by using a simple equation. For people who aren't too keen on math, don't worry, it's pretty simple.

To work out our expected value we multiply the potential gain by the probability of that gain occurring (for example saving $5 by not paying the parking ticket by 90%).

Then we must find the product of the potential loss and the probability that the loss will occur and minus this value from the first part.

We have all this information from the real life example above:

Potential gain = $5, Probability of that gain = 100%-10% = 90%.

Potential loss = -$60, Probability of that loss = 10%

The wording may be quite confusing but the expected value formula will make more sense.

Expected Value Formula

Expected value, EV = (probability of gain)*(value of gain) + (probability of loss)*(value of loss)

For our parking ticket example this becomes:

EV = (0.90)*($5) + (0.10)*(-$60) = $4.5 – $6 = -$1.5

So on average, every time we don't pay our parking ticket we will stand to lose $1.5.

This may confuse people as in no single case can we lose $1.5 – we either save $5 or we have to pay $60.

The $1.5 comes from is the average loss we will make over a long period of time. So if we did this 100 days in a row, 90 of the days we would have saved $5 each day for $450, but we would have been fined 10 times for $600 total.

Thus on average, we would have lost $150 which is $1.5 per day.

Change Variables and Find The Breakeven Expected Value Point

Using this simple calculation we can see how changing the variables affects our expected value; if it is less likely we will be caught and fined we should not pay for the ticket, that is obvious. For example, it is 1% likely that we will be caught without a ticket:

EV = (0.99)*($5) + (0.01)*(-$60) = $4.95 – $0.6 = $4.35. Therefore it makes sense to not buy a ticket for these parameters.

Intuitively that makes sense too since it is so unlikely we will have to pay $60 dollars.

If the fine cost less or the cost of parking cost more, that will increase the EV of not buying a ticket.

We can also find the point at which it becomes profitable to start not paying for tickets.

We find the breakeven point by setting the EV to zero and then find the probability that we will be caught which we have labelled as X.

This is called the breakeven point (requires some algebra):

EV = 0 = (x)*($5) + (1-x)*(-$60) => 5x-60 + 60x = 0

Therefore: 65x =60 and x = 60/65 = 92.3%.

So if we are likely to be caught greater 7.7% of the time we should buy a ticket; and if less than that we should not buy a ticket.

We can also apply this analysis to poker in situations where we know our equity pot odds, bet size and pot size.

Expected Value In Poker When Calling

To go back to the previous hand example with A9 of diamonds we had pot odds of 28% and we had pot equity of 18%. The pot odds are based on how much he bet and pot equity is based on the hands we assumed he would be bluffing with. So let’s put that all together to do an expected value calculation.

Again the expected value will be:

EV = (probability of gain)*(value of gain) + (probability of loss)*(value of loss)

The probability of gain will be our equity which is 18%.

The probability of loss will be 1-equity which is 82%.

The value of our gain will be $67.5 and the value of our loss is -$26. Subbing in:

EV = (0.18)*(67.5) + (0.82)*(-$26) = 12.15 – 21.32 = -$9.17

Therefore this will be an unprofitable call to make. That should also be intuitive since we are winning so infrequently.

Expected Value When Betting

Here is another example of when betting on the river.

We opened to 3bb from UTG and got one call from the Button. On the flop of 9h7s3s, we have an inside straight draw and two overcards so we decide to continuation bet the flop and 3c turn.

Now on the river, we have to decide whether to bet again.

Firstly we can assume that we will lose if we check this river. We only have Jack high and unless our opponent has a busted draw AND checks, we will not take this pot down.

Therefore, checking has an EV of close to zero.

So what is the expected value of betting?

Well, we need to know:

• Our bet size = $32
• The pot size before our bet = $42.5
• How often our opponent calls = ?

How often our opponent calls is dependent on a lot of factors, but taking a rough estimate, let's say he will fold 50% of the time since we have shown a lot of strength and the river is a scare card.

We then the expected value equation becomes:

EV = (probability of gain)*(value of gain) + (probability of loss)*(value of loss)

EV = (0.5)*($42.5) + (1-0.5)*(-$32) = $5.25

Therefore, betting $32 is going to be very profitable in this spot if our opponent folds 50% of the time – the reason for that: our opponent is folding too much.

He should be calling at least 57% of the time to prevent us from making an immediate profit according to the minimum defence frequency formula which we will cover in the next section.

EV = (0.43)*($42.5) + (1-0.43)*(-$32) = ~$0

This J♦T♦ example is quite simple because we are on the river and there is only one more betting round to play.

However, it gets way more complicated when trying to estimate EV on the flop or turn for a number of reasons:

• You don't know what cards will come on the turn or river
• You don't know how your opponent will react to each of these cards
• There are so many different ways the hand can be played out, it's almost impossible to work out each path by hand.

For that reason, simple EV calcs like this are usually only performed on the river and software such as Cardrunners EV is used to simulate multiple street EV bets. Although these simulations take a lot of time to set up.

Minimum Defense Frequency

As mentioned, the minimum defence frequency is how often our opponent should call our bet to prevent us from making an immediate profit.

Poker Starting Hand Chart

We can work our opponent's minimum defense frequency using this formula:

MDF = pot size / (pot size + bet size)

For the last example:

MDF = $42.5 / ($42.5 +$32) = 0.57 or 57%

Notice that increasing our bet size, decreases how often our opponent should call.

And the opposite is true too: decreasing our bet size increases how often our opponent should call.

Poker Value Hands

A Quick Trick To Determine Profitability When Calling

A quick way to determine the profitability of a call without doing a full EV calculation is to compare the pot odds and our equity.

In the case of the example hand, we have pot odds of 28% and we have equity of 18%. If the equity is less than the pot odds we should not call as it would be unprofitable; conversely if we had more than 28% equity we can make the call as it would be profitable to do so.

You are not expected to do these calculations in their head while at the poker table. The purpose of the calculations is to analyse difficult hands off the table after your poker session, not during a game. This allows you to determine if you made the right decision and correct mistakes.

One of the main drawbacks of EV calculations is that you cannot perform them before the river as there are so many variables at play. We do not know what our opponent will do on future streets, will he check or will he bet? What card will come on the turn and river? If he does bet, what size will he use?

This is what makes poker such a complex game and a difficult one to solve computationally.

Despite this fact, expected value, pot odds and equity are useful in both poker and real life. So it is worth taking the time to understand how it is applied.

Expected Value Calculator

Poker Expected Value Calculator

Make sure you check out this expected value calculator over at RedChip Poker:

It is extremely easy to use. All you have to type in are the three values in the above fields and it returns the expected value of that particular situation.

Poker Expected Value Starting Hands Meaning

Here is a video which recaps the main points we covered in this lesson:

Closing Words

So that's it for our lesson on Expected Value. You should have already worked through the counting outs, pot equity and pot odds lessons before doing this so if not make sure you check them out.

You should now have a much better understanding of the math behind poker – your next step is to put it into practice and perfect it. Good luck!

Head back to poker 101 to learn more or check out the blog page for blog updates.