Explanation of Expected Value
This concept is a mathematical expression that represents the anticipated average outcome of a particular situation in poker.
Example usage → “The expected value of this call on the river in the long run is approximately 5 big blinds.”
Let’s explore how the formula can be demonstrated using an example from Texas Hold'em.
On the turn, we face an all-in wager of $50 against a $100 pot. We anticipate having around 40% equity in the pot relative to our opponent’s possible hands. So, what is the expected value of our call?
To begin, we need to outline the four components that contribute to the formula for calculating expected value .
Probability-of-winning – 40%
Probability-of-losing – 60%
Winning amount - $150 (this includes the pot and the opponent's bet).
Lose-amount - $50 (Our investment)
Now, let's substitute these components into the EV formula. Remember that percentages must be converted into decimal form (by dividing by 100).
(0.4 * $150) – (0.6 * $50) = EV
While working with brackets in calculations, it’s crucial to first address the values inside the brackets before proceeding with other mathematical tasks.
($60) – ($30) = $30
From this, we can expect to earn an average of $30 each time we make this call. This aligns with our pot odds; as long as we have over 25% equity, our call is deemed profitable.
Advanced EV Calculations
The example above illustrates a straightforward calculation of expected value. This simplicity stems from at least two key factors:
1) We don't need to consider additional factors like fold equity.
2) Our action is an all-in call, so we don't have to deal with intricate variables on the river.
Now, let’s examine a slightly more intricate scenario where we have to take both pot equity and fold equity into account.
In this situation, there’s $100 in the pot on the turn. We go for a semi-bluff, going all-in for $50. We expect to have 20% equity when called, and we anticipate our opponent will fold roughly 30% of the time to our bet. What is our expected value here?
One effective approach for resolving such scenarios is to dissect them into all possible outcomes along with their corresponding gains or losses.
Event A – The opponent folds, and we win the $100 pot. (This occurs 30% of the time).
Event B – The opponent calls, and we take down $150. (This happens (0.7*0.2) 14% of the time).
It's important to note that to calculate the probability of successive events (for instance, the opponent calls, and then we win), we multiply their probabilities together (after converting to decimal). Our opponent calls 70% of the time when we bet, and we will hit our draw 20% of the time. Thus:
0.7 * 0.4 = 0.14 = 14%
Event C – The opponent calls, and we lose our $50. (This occurs (0.7 * 0.8) 56% of the time).
The fundamental EV formula can be adapted to include a separate section for each event.
Let’s input the details regarding our three events into a three-part formula.
(0.3 * $100) + (0.14 * $150) + (0.56 * $-50) = EV
Pay attention that the section representing event-C incorporates a negative number to indicate losses.
($30) + ($21) – ($28) = $23
Even with less equity than our first example, our turn semi-bluff still yields a positive expected value of $23. This highlights the advantage of having fold equity when we opt to play our draws aggressively instead of passively. If we were instead facing a bet on the turn rather than making one, we wouldn’t be getting adequate odds for a +EV call.
Tree Building Software
The complexity level discussed here is likely the maximum that individuals can calculate manually. It's not that people cannot handle more intricate situations; rather, it’s a matter of feasibility. Performing extensive calculations by hand is impractical when we can leverage computer software to accurately compute expected values for complex game trees.
The most widely used software available for calculating expected value in complicated game trees is CardrunnersEV. (A game tree is simply a model that illustrates all potential actions that can be taken in a poker hand, and these can become quite extensive quickly).
Example of Expected Value in context -> (Hold’em) Holding Aces before the flop signifies that this decision surely possesses a positive expected value .
Integrating Expected Value in Your Poker Strategy
A thorough grasp of expected value is crucial for advanced play. Skilled players not only recognize which moves are inclined to hold positive expected values , but they can also provide a general assessment of how lucrative those moves are likely to be. In contrast, novice players might be able to identify promising moves but typically struggle to calculate precise EV.
Let's consider a straightforward example where the precise EV should be evident without needing complex calculations.
The opponent goes all-in for $50 into a $100 pot on the turn. We estimate our equity at 25% and decide to call. Calculate the following -
a) The expected value of a call.
b) The expected value of a fold.
c) Assess the expected value for the entire hand.
a) We are receiving the exact pot odds necessary given that our pot equity stands at 25%. Therefore, our expectation will be exactly 0.
b) Folding will always have a zero expectation. It’s essential to understand that this expectation of 0 relates only to the current decision point. This doesn’t imply we break even over the whole hand.
c) Many players might mistakenly conclude that our expectation is 0, simply because it was a break-even call with our draw. This view reflects our relative expectation but fails to capture our expectation for the entire hand (absolute expectation).
The expectation for our overall hand relies on how much we have already contributed to the pot. With $100 already in the center and us being in a heads-up situation, it's reasonable to estimate that we have invested about $50. Consequently, our overall expectation for the hand is -$50 (a loss of $50).
Relative vs Absolute Expectation
Differentiating between relative and absolute expectation is crucial when discussing expected value . It might come as a surprise to many players that making a “break-even” call with a draw could lead to an overall loss for that hand. This doesn’t indicate that the call was an error, but rather that the turn cards favored our opponent, resulting in a loss. While we have some control over our relative expectation, our absolute expectation often merely reflects how we are performing during a specific hand.
Max-EV vs +EV
The ability to make rough assessments of our EV is valuable because it enables us to compare multiple profitable strategies. In poker, it’s entirely possible for more than one approach to yield a positive expectation. In such cases, an astute player will always opt for the strategy that maximizes expected value , a practice referred to in the industry as “max-EV”.
See Also
Pot Odds , Expectation , Equity , Turn , Call , Actions , Semi-bluff