In part one of the Expected Value Calculations series, we looked at a simple method for calculating the expected value. We only looked at situations which represent calling an all-in bet. If we are the one betting our calculations will change slightly . However they will still derived from exactly the same formula.
When you are calling an all-in bet, you are only concerned with your pot-equity. When you are the one betting you are also concerned with something else; fold equity. Fold-equity can be defined as additional equity in a hand given your opponent may fold.
Sometimes we will find ourselves in situations where we have no pot-equity (our hand is garbage), and the only thing going for us is our fold-equity. These situations are known as pure-bluffs and will be the subject of this article.
Let’s take a look at an example -
You are in position on the river with 6high. There is $100 in the pot. You make a bluff of $60. You estimate your opponent will fold about 50% of the time. What is the expected value of this bluff?
(If you think you can do this already, have a go and then compare your answer to the one below)
Firstly let’s remind ourselves of the EV formula from the first article:
The difference is that rather than considering our pot-equity to obtain values of P(winning) and P(losing) we will consider our fold-equity. We “win” when our opponent folds. If our opponent is folding 50% of the time P(winning) = 50% or 0.5. The P(losing) must then equal 100% - 50% = 50% or 0.5. Losing and winning is equally likely in this case. Win-amount should be straightforward, it is the amount contained in the pot $100. The lose-amount is the amount we are investing in our bluff; $60.
Let’s input those numbers into the EV formula
(0.5 x $100) – (0.5 x $60) =
$50 - $30 = $20
This bluff has an expectation of $20. Whenever we bluff in this spot we make $20 on average against this opponent.
Calculating “Break-Even” Point
Let’s take the same situation but consider it in a different way. It is useful to be able to calculate what is known as the “break-even” point of a bluff. Rather than calculate the expected value of a bluff, we look to see how often a bluff needs to work in order to break even.
You are in position on the river with 6 high. There is $100 in the pot. You make a bluff of $60. What is the break-even point of this bluff? (In other words, how often does our opponent need to fold for us to have a neutral expectation?)
We can use the formula for EV to calculate this by inputting the values we know and letting them equal 0. This will be demonstrated shortly; but there is actually a much simpler method for calculating break-even point of a bluff which we will look at first.
Simple huh? The percentage of the total pot we are contributing is identical to the percentage of the time we need our opponent folding in order to break-even. How does this apply to our question?
What percentage of the total pot are we investing? After our $60 has gone into the middle the total pot will be $160. We can express this as the fraction 60/160. Hopefully you recall from part 1 how do convert this into a percentage. 60 ÷ 160 = 0.375 = 37.5%.
The break-even point of this bluff is 37.5%. If our opponent folds more than this our bluff will have a positive EV. If he folds less, then our bluff will not be profitable. The more chips we risk on our bluff, the more often we need our opponent to fold in order to show a profit.
Let’s see how we can use the formula for EV to obtain the same result. Feel free to skip this step – it is not mandatory and will require knowledge of basic algebra. It’s interesting to see how we can derive the calculation for break-even point from the formula for EV. All EV calculations however complex, will stem from the same basic formula.
We will use X to represent the decimal chance of our opponent folding. We want to find the value of X when our EV is 0.
(x X $100) – ((1- x) x $60) = 0 (P(losing is always “1 – P(winning)” as a decimal)
$100x - ($60 - $60x) = 0 (Simplify the brackets)
$100x - $60 + $60x = 0 (Remove the brackets. 2 “-“ signs make a “+”)
$160x = $60 (Arrange to have β on one side)
x = $60 ÷ $160 (Divide both sides by $160)
x = 0.375
Take a look at the following questions – and perhaps try to find an answer.
You are in position on the river with 6high. There is $200 in the pot. You make a bluff of $110. You are facing two opponents. What is the break-even point of this bluff?
This one should be pretty straight forward.
110 ÷ 310 = ~ 0.355
So we need our opponents to fold a combined 35.5% of the time in order for this bluff to be profitable.
You think one opponent will fold 80% of the time and the other 50% of the time? What is the expected value of this bluff?
The difference is now that we have 2 opponents and need to know the combined percentage of the time they are folding. We need to learn a specific rule of probability theory to answer this.
In simple terms – to find the probability of 2 events occurring one after another you multiply their probabilities together. They will need to be in decimal format for you to do this. We want to find the probability of one player folding followed by the other player folding. We multiply the respective probabilities –
0.8 x 0.5 = 0.4 = 40%
To calculate the exact EV of this bluff however –
(0.4 x $200) – (0.6 x $110) =
($80) – ($66) = $14
Calculating the probability of successive events is useful also. For example what are the chances of getting dealt AA twice in a row? The chance of getting dealt AA is roughly 0.45%. The probability of getting AA twice in a row is therefore –
0.0045 x 0.0045 = 0.00002025 or 0.002% or 1/50000
It doesn’t just work for 2 events. The probability of losing 5 coin-flips in a row is
0.5 x 0.5 x 0.5 x 0.5 x 0.5 This can be expressed as (0.5)^5 (0.5 “to the power of” 5)
= 0.031 or 3.1%
It’s all very well using a percentage representing how often our opponents folds, but how do we know how often he folds. The simple answer is, we don’t. All we can do is make an estimate based on:
- Our opponents’ tendencies
- Our opponents’ ranges
- Our range
- Our bet-sizing
This is the reason why software such as HEM or PT cannot calculate your true expected value. They cannot make estimates about your opponent’s range or the likelihood of them folding to various bet sizes. What they can do calculate is “allin-EV”. We looked at an example of allin-EV in part one; we simply called an all-in bet with a certain amount of equity.
When it comes to the EV of a bluff, an estimation of your fold-equity will be required. This means your EV calculation will only be as accurate as your estimate of your fold equity. Estimating fold equity in various situations is hence an important skill if you want to determine the expectation of various bluffs.
Read more Expected Value Calculations articles