Advanced Expected Value
Before you get scared that this poker lesson is all about complex math, it’s not. Yes we’ll cover some of the calculations used to determine expected value in common poker situations, but EV is just as much about a mindset as it is math. Instead of just thinking "I’m ahead" or "I’m behind", the EV mindset is about putting your opponent(s) on a range of possible hands and developing a habit of quickly analyzing each possible decision and picking the one that will make you the most money in the long run.
At this point in your poker training, it’s assumed that you understand basic poker pot odds and hand strength. If you are not confident in these aspects, please go back and re-read the beginner level lesson “An Introduction to Pot Odds” and the follow up advanced lesson, “Perfecting Pot Odds“. As we take your understanding of expected value to another level, there are additional criteria that must be incorporated that will affect the long-term profitability of your decisions. The following are the primary criteria you will use in thinking about expected value:
- What is my opponents likely hand range given my reads and the action thus far? And:
- How likely is my opponent to fold if I bet?
- How likely is my opponent going to bet in this situation?
- How likely is my opponent bluffing in this situation? Value betting?
- How likely will my opponent call with a hand that I am beating?
- How likely will my opponent fold a better hand if I bet?
- What will my stack size be as a result of my decision?
- What are the odds of improving my hand with cards to come?
- What are my implied odds if I hit the card I am looking for?
Common Advanced EV Situations
One of the most common poker scenarios where knowledge of expected value comes into play is when you’re short stacked and are faced with a decision to shove all-in or fold. Calculating your fold equity and pre-flop ranges of your opponents is crucial. Put simply, the factors that will dictate your decision include:
- How much you win in blinds and antes if you are not called.
- How often you’re going to get called.
- How wide your opponents calling range is.
- Your hand and how it fares against your opponents range.
- How much you gain if you get called and win.
- How much you lose if you get called and lose.
Your image, your opponents’ images, your stack size and the stack sizes of the players left to act will play a big role in your opponents calling range. A very large stack size might decide to gamble more to bust you out, while a very short stack might feel desperate to double up; both of which may have a larger calling range. Of course a looser player will have a wider calling range, where a tight player may only call with premium hands.
Tools to Calculate Expected Value
In a moment we’re going to look at some sample hands in which utilizing an advanced understanding of expected value will help you earn more profit over time. First, let’s mention a couple of useful tools that which will help you to analyze your hands and determine the maximum EV of each situation:
- Poker Stove – This is arguably the most important as it will automatically calculate how far ahead or behind you are based on your opponents range of hands. You can download it for free at PokerStove.com
- SNG Power Tools – This software makes it easy to run an EV calculation that takes into account the ICM, folding equity, and more. Click here to visit SNG Power Tools.
Hand Example – Blind vs. Blind
- No Limit Hold’em Tournament Play
- Blinds: 150/300
- You have 2,000 in the SB, the BB has 3,500.
Reads
You have been playing fairly tight and have not made any recent moves. You are in the small blind. The big blind seems to be a fairly tight player, but also understands short stack dynamics.
Pre-Flop
It folds all the way to you and you hold Q
8
in the small blind:
Figure 1
Should you fold, call or shove? To simplify the example, let’s assume calling is not an option (since you are short stacked, it’s not an option anyway). To answer this, we need to assign a calling range to the big blind. Based on our read, it is fair to assume they would call with pocket pairs of 77+ and A7+. This comprises about 12% of all possible hand combinations. So we need to know how well we fare against his range if we do get called. To calculate this, we’ll use Poker Stove:
31.275% { Q8 }
68.725% { 77+, A7s+, A7o+ }
Obviously we are behind here, but our opponent is only calling with about 12% of all hands. This means that the BB folds 88% of the time. If we do get called and win, our stack will double to $4,000, and of course we’ll be at $0 if we lose.
Based on these calculations, here are all the possible outcomes:
- We fold: after posting the SB, our stack is now 1,850.
- We shove, BB folds: stack is 2,300 x 88% likelihood = $2024 expectation.
- We shove, BB calls, we win (12% calling X 31.275% winning = Occurs 3.753%): 4,000 x 3.753% = $150.12 expectation.
- We shove, BB calls, we lose (12% calling x 68.725% = Occurs 8.247%): $0 expectation.
So if you add up all the possible outcomes for shoving, you get $2024 + $150.12 + $0 = $2174.12.
+2174.12: Total expectation of shoving
– 1850.00: Total expectation of folding
$324.12: Total expectation
As you can see, you gain an average of +$324.12 EV in the long run if you push all-in. But we just used a hand that was statistically better than an average hand the BB could hold. What if we pushed all-in with a trash hand, like 72os?
24.931% { 7
2
}
75.069% { 77+, A7s+, A7o+ }
- We shove, BB folds: our stack is 2,300 x 88% = $2024 expectation.
- We shove, BB calls, we win (12% calling X 24.931% winning = Occurs 2.99%): 4,000 x 2.99% = $119.60 expectation
- We shove, BB calls, we lose (12% calling x 75.069% = Occurs 9.01%): $0 Expectation
+2143.60: Total expectation of shoving (2024 + 119.60)
– 1850.00: Total expectation of folding
$293.60: Total expectation
So because our opponent is only calling with 12% of hands, our shove with the worst hand in poker is still highly profitable. You can play around with Poker Stove and change your opponent’s possible calling range to determine various expectations.
Hand Example – Hero Call on the River
- No Limit Hold’em Tournament Play
- Blinds: 100/200 + 25 Antes.
- Both you and your opponent started the hand with 5,000.
Reads
The big blind has shown to be a loose player capable of calling people down with marginal hands and chasing draws. He has also played his bigger hands strong.
Pre-Flop / the Flop
From the button, you raised pre-flop to 500 with TT and the big blind defended. On a flop of A67 your opponent checks and you bet 850 into a pot of 1325:
Figure 2
Your opponent calls your 850 bet. Now let’s look at hand ranges. Since your opponent is loose and likes to see the flop, his defending range pre-flop is pretty wide. Since he would most likely re-raise with AQ+ and 88+, I would assign a range of A2-AJ, K9+, Q9+, 22-77 and all connectors and one-gappers 56 – JT. Since he has called our continuation bet, we can narrow down his range. Based on our read of him playing bigger hands strong, we can eliminate AT, AJ, 66, 77, 67, A6 and A7. Removing those and all other hands that did not connect with this flop, we are down to A2-A5, A8,A9, 89, 56, 57, 97, 87, 86 and all flush draw combos. It’s possible he may have check-raised with a strong draw, but our read on this player is that he usually calls in these situations.
The Turn & River
The turn card brings a 2. Based on our read, this won’t change much unless our opponent turned two-pair with A2 or hit a set of 2’s. Our opponent checks and we check behind for pot control. The river is K and our opponent shoves all-in for 3,650:
Figure 3
The pot is now 3,025 and our stack is also 3,650. The king is a tricky card because we are losing to any Ace and any suited King that was chasing a draw, but we are way ahead if our opponent was on a draw or had middle pair. Our opponent could have two-pair or a set, but based on our original read of his range this isn’t likely, however we cannot rule it out completely. So let’s look at the EV of the situation.
Pot Odds: 3025 in the pot + 3650 in his stack = 6675 and it’s 3650 to call. This is 6675/3650 or 1.83 to 1. So we cannot be behind more than 1.83-1 to make this call profitable.
53.125% { TT }
46.875% { 55-22, A9s-A8s, A5s-A2s, KsQs, KsJs, KsTs, Ks9s, QsJs, QsTs, Qs9s, JsTs, Js9s, Ts9s, Ts8s, 9s8s, 9s7s, 8s7s, 8s6s, 75s, 65s, A9o-A8o, A5o-A2o, 97o+, 86o+, 75o, 65o }
Using Poker Stove, you can see that we are actually slightly ahead by 1.13 to 1. Simply based on pot odds, this would be a +EV call to make. But sometimes we need more confidence in iffy situations. Let’s see what our actual expected value is.
To calculate our expected value, here are all the possible outcomes:
- We fold: Stack is 3,650
- We call and win (53.125% of time) = 10,325 x 53.125% = $5485.16 expectation
- We call and lose (46.875% of time): $0 Expectation
+5,485.16: Total expectation of calling
– 3,650.00: Total expectation of folding
$1,835.16: Total expectation
As you can see, by making this call you will win an average of $1,835.16 over time, giving it a high positive expectation.
ICM (Independent Chip Modelling)
ICM determines your equity share of the prize pool in a tournament based on the stack sizes of the remaining opponents and the probability of your finish. With these probabilities, a dollar value can be associated with your stack size. For example, if the total prize pool remaining is $1,000 and you have 50% of the chips in play, your equity is $500 (assuming the players remaining are around the same skill level).
What does this have to do with expected value? Well, in cash games the expected value of your chips (cEV) equals the same amount as the real dollar value associated with them ($EV). In the beginning parts and middle stages of most tournaments cEV roughly equals $EV, however depending on the payout structure and how many players are remaining, there are times in which a decision can be +cEV but -$EV. This is usually much more prevalent in SNGs, but can also apply when there are large payout jumps at the final table of a tournament. ICM helps you determine your $EV in these situations. For more information about ICM and to play around with an ICM calculator, visit: www.chillin411.com/icmcalc.php
As a conclusion, the best way to train yourself to consistently make +EV plays is to analyze key hands after each session of play. This will allow you to assign hand ranges with a clear head and make an unemotional assessment as to whether a play had positive or negative expectation. The more you train yourself to become analytical with these decisions, the more profit you will make at the poker tables.
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