Expected Value in Texas Holdem

An important concept that most winning Texas holdem players understand is expected value.

The expected value is the average amount you win or lose on a situation if you were able to play the exact same situation thousands of times.

It can be difficult to understand expected value on a hand for hand basis, but if you ran a situation 100 times it can help make it clear.

Here's an example:

You're finished with the betting round on the turn and are waiting for the river card to be dealt. You have four cards to a flush and if you complete the flush you'll win the hand and if you don't complete your flush you'll lose the hand. Nine out of the 46 remaining unseen cards win the hand for you and there's $100 in the pot.

The percentages say you'll win the hand 19.57% of the time. You can figure the percentage yourself by dividing nine by 46.

If you play the exact same situation 100 times you win 20 times and lose 80 times. We rounded the 19.57% up to 20.

So 20 times you win $100 for a total win of $2,000. If you divide $2,000 by 100 times you end up with the average win, or expected value for this situation. In this case the expected value is $20.

This is a simplified example and isn't especially useful at the holdem tables. But if we take the reasoning and mathematics behind what you just learned a bit deeper you can find a way expected value can be quite valuable and useful at the Texas holdem tables.

If you take this example to the next level consider this situation.

The cost of the call, $20, is multiplied by 100 to come up with a total cost of $2,000 to play the situation 100 times. The 20 times you win the hand you win $100. 20 times $100 is $2,000. So it looks like your expected value is 0 in this situation.

But there's still one thing to consider. What happens on the river when you miss your hand and when you hit your hand? If you don't check and fold on the river every single time you miss your hand your expected value goes below even.

Will your opponent ever call a bet on the river if you hit your flush? The answer is certainly yes. They might not call often, but you can get action on the river with a flush. This actually pushes the expected value of a hand like this to the positive side.

As a Texas holdem player you need to make it your goal to find as many positive expectation situations as possible and play in every one of them possible. You also need to avoid negative expected value situations like the plague.

The magic of positive expectation is the short term results don't mean anything. If you consistently put yourself in positive expectation situations you'll win money in the long run.

Statistical laws show you have to make money in the long run if you always play in positive expectation situations.

Here's a list of a few positive expectation situations:

Expected value is often shortened to EV. You may see positive expected value listed as +EV or negative expected value listed as –EV.

One of the biggest mistakes Texas holdem players make when trying to wrap their head around expected value is trying to figure out how the money they've already placed in the pot gets figured into the equation.

The answer is simple, but most players have a hard time with it. The money you've already put in the pot is only considered in the pot size. In other words, the money stops being yours as soon as it goes in the pot.

Examples of Expected Value

The best way to learn how to determine expected value in Texas holdem is to practice. This section includes many examples so you can practice for free. When you practice at the tables it can cost you money.

Take a few minutes and try to figure out the correct answer before looking at the solution. Remember to run the situation as if it was identical 100 times. Just follow the simple steps used in the opening section.

The examples all come first and the solutions are further down the section. This way you can't cheat to see the answers before you try to figure out the answers unless you want to. All of the examples are using Texas holdem.

On the river of a no limit game you have the top pair with a good kicker but only think you have a 20% chance of having the winning hand. The pot has $500 in it, you check, and your only opponent bets $250.

You're playing a $10 / $20 limit game and after the turn you have an open end straight draw and a flush draw. The pot has $100 in it, you check and your opponent bets $20.

If you raise your opponent will call on the turn and call one bet on the river if you hit your straight, but will fold to a bet if you hit your flush. If you miss your draws you check and fold to a bet on the river.

On the river of a no limit game the pot has $2,000 in it and you just hit a full house on a board that has three suited cards. The way the hand played out you're relatively sure your opponent hit the flush. You have to act first and are trying to determine the best way to extract the maximum expected value from the situation.

You can check and raise if your opponent bets or you can bet. The mounts of bets and raises complicate the situation, but being a winning Texas holdem player is complicated, so you have to make your best educated guess when situations like this come up.

Based on what you know about your opponent if you make a bet up to $2,000 she'll call. If you check she'll bet $500 and call up to a re-raise of $1,000.

You're playing in a $20 / $40 limit game and flop an open end straight draw. The pot has $80 in it at the start of the round, the first player bets, the second folds, the third calls, and you're last to act. The pot now has $120 in it and you have to call a $20 bet to see the turn.

After the river has been dealt you have top pair and top kicker. You determine you have a 40% chance of winning the hand because the way the hand has played out your opponent either has top pair with a worse kicker or hit two pair. Your opponent has played the hand aggressively enough that you've tilted the percentage to her favor.

The pot has $1,000 in it before your opponent bets $800. Once you know the break-even expected value it's easy to see if a call or fold is more profitable in the long run.

If your percentage is correct what's your expected value if you call?

How much would your opponent have to bet to make your call a break even expected value?

If you call $250 100 times your total investment is $25,000. The total amount of the pot is $1,000 after you call. Winning 20% of the time means you win a total of $20,000 when you win. This is a negative expected value of $5,000 total and $50 on average.

You need to win this hand at least 25% of the time to break even. You know this because the total investment stays the same, creating a total amount of $25,000. You divide this by the size of the pot to find the break-even point. $25,000 divided by $1,000 is 25, so you need to win 25 out of 100 times, or 25%.

This situation has a host of possibilities so you need to consider them one at a time. Before moving deeper you need to decide if a fold or call is correct.

You're faced with a call of $20 making a total pot of $140. You have 15 outs out of 46 unseen cards for a percentage of 33% chance to win. Your total investment over 100 hands is $2,000 and the 33 hands you win return $4,620. This creates an average positive EV of $26.20 per hand. So you can rule out a fold.

Now let's consider a raise. Three things can happen if you raise, so you need to consider each of them and then combine the results.

The first thing that can happen is you raise, your opponent calls, and you miss your draws. Your raise costs $40 so over 100 hands you lose $4,000, or $40 on average. This happens 31 times out of every 46 possibilities, or 67 times out of 100.

The second possibility is you raise, your opponent calls, you hit a flush, and you don't win additional money on the river. Over 100 hands your raise still costs $40, making a total pot of $180. You win $180 100 times for a total win of $18,000. When you subtract your investment of $4,000 you have a positive expectation of $14,000. This is an average of $140 per hand. You hit your flush 20 out of 100 hands.

The third possibility is you hit your straight. In this case you bet $40 on the turn and another $20 on the river for a total investment over 100 hands of $6,000. The total pot size after all betting on the river is $220, for a total win of $22,000. This is an average win of $160 per hand. You hit your straight and not a flush 13 out of 100 hands.

When you combine the results you have the following:

• 67 times out of 100 you lose $40.
• 20 times out of 100 you hit your flush and win $140.
• 13 times out of 100 you hit your straight and win $160
• 67 times 40 = a loss of $2,680
• 20 times $140 = a win of $2,800
• 13 times $160 = a win of $2,080

This makes a total positive expected value of $2,200, creating an average of a $22 +EV per hand.

When you compare this to the +EV of $26.20 per hand created by calling it shows both options are profitable but a call is correct in this situation.

Realize that if you can extract more money on the river than in this example a raise may increase to a point where it has the higher EV.

In the first situation a bet of $2,000 in 100 hands is a total investment of $200,000. The total pot size with your opponents call is $6,000, for a total win over 100 hands of $600,000. This is a positive expectation of $400,000 over 100 hands for an average of $4,000.

The second situation requires a total bet of $1,500, covering the $500 bet and the $1,000 raise. This makes a total investment of $150,000 over 100 hands. The total pot size is $5,000 so the total win over 100 hands is $500,000. This creates an expected average value of $3,500.

So the correct play is to bet $2,000.

This may seem like a simplified example, but this is a perfect example of the complicated situations you fin at the holdem tables on a regular basis. When you start considering all of the possible outcomes for each hand being able to determine expected value goes a long way to maximizing your long term profit.

The first thing to determine is the expected value from the flop to the turn. You've seen five cards so the deck has 47 unseen cards and eight of them complete your straight. This means that 17% of the time you'll complete your straight on the turn.

It costs you $2,000 to call the $20 bet 100 times and the 17 times you win the total amount won will be $2,380, assuming no further action in the hand.

But the odds of no further action taking place in the hand are slim. Also, what happens if you miss your draw on the flop?

Unless the expected value is close to even you don't need to determine how likely you'll get additional action is when you hit. If the EV is close to even or slightly negative the expected future action is enough to push the percentages to make a call correct. That's all you need to know to continue with the hand based on possible future action.

The next thing to consider is what happens when you miss your draw on the turn. The pot is now $140 and the bets are $40. The only way you'd ever consider folding in this situation is if you get caught in a bidding war between the other two opponents, and even then with capped betting rounds the expected value says to call.

More likely you'll face a single bet or two bets at most. The first thing you need to do is determine if the situation still offers a positive expectation if you face two bets.

Two bets from each of your opponents make the pot $300 and you have to call $80, making a total pot size of $380.

You've now seen six cards, leaving 46 unseen and you still have eight outs. Your percentage chance of winning has improved slightly but it still rounds down to 17%.

Your total cost to call 100 times is $8,000. The 17 times you win you get $380, for a total win of $6,460, creating a negative expectation situation of $15.40 on average.

This is where you need to make a judgment call based on how much you think you can extract from your opponents on the river when you hit your hand. You need to win an average of $470 total instead of the $380 listed above to break even, so can you get over two additional bets on the river when you hit?

An open end straight draw is harder to see when it hits for your opponents than a flush, and you're in good position, so you can probably push your wins enough when you hit to make this a break even play or a slightly positive EV play. But it's close, so it really helps to know your opponents.

What about if you only face a single bet from each of your opponents?

In this case you have to call a $40 bet and the size of the pot is $260 with both opponent's bets and your call. It costs $4,000 for 100 calls and the 17 times you win the total amount is $4,420. This is a positive expected value and is a clear calling situation. You'll actually win more when you hit your hand in most situations from action on the river.

The last thing to think about is if you should actually raise on the flop.

If you raise what will your opponents do? To get a true picture you need to run every possible situation, but for the sake of this discussion let's assume one opponent folds and the other calls.

The pot has $120 in it, you raise $40, and the remaining opponent calls $20 for a total pot of $180. Your raise in 100 hands totals $4,000 and you still win 17 times. 17 times $180 is only $3,060, creating a negative expectation situation.

When you factor in the possibility of both opponents folding and winning more bets on the turn and river when you hit it still isn't enough to make a raise enough. Remember that sometimes your opponent will re-raise, making the situation worse.

This is a complicated example so if you don't understand all of it, take the time to go back over it and study it. None of the calculations are overly complicated, but it can be confusing when you run into so many of them.

It's going to cost you $800 to call, so you multiply that by 100. So your total cost is $80,000. The 40 times out of 100 that you win you'll win a pot of $2,600. 40 times $2,600 is $104,000. So the total amount of your wins minus the cost of making the call is $24,000. If you divide this by 100 your average expected value is $240 every time you're in this situation.

To determine the break even amount your opponent would need to bet requires a slightly different calculation. Your opponent would need to bet $2,000 to create a situation where your expected value is zero.

A bet of $2,000 costs $200,000 to call 100 times. The pot is $5,000, so when you win 40 out of 100 times you win a total of $200,000, creating a zero expected value.

This means that any bet below $2,000 in this situation has a positive EV to call.

More importantly, consider how important it can be to call almost every bet on the river if you have a 40% chance to win. You can work these numbers for any percentage chance of winning to determine if a situation offers positive or negative EV. Most players fold too often to small and medium bets on the river.

You can use a complicated mathematical formula to determine this amount, but it's simpler for 99% of the population to do a simple progression of possibilities.

Here's exactly how we determined that a $2,000 bet is the break-even point.

We know that a bet of $800 creates a large positive expectation situation so a break-even will need to be quite a bit larger than that. So we built a small table and started plugging in bets.

Bet Amount	Total Pot	Call X100	40 Wins X Pot	Average EV
$1,000	$3,000	$100,000	$120,000	$200
$1,500	$4,000	$150,000	$160,000	$100
$2,000	$5,000	$200,000	$200,000	$0

Don't be scared or intimidated by these calculations. Once you do a few of them you'll quickly learn they aren't too difficult. Pick a different situation and build a table to find the correct break-even point.

You need to practice these quite a bit so you learn to closely approximate your expected value at the table. It's difficult to determine all of this in your head, but as you gain experience you'll learn to recognize profitable and unprofitable situations.