Thus far we have considered Expected Value calculations in 2 scenarios.
- In the first article we calculated the expectation of calling an all-in bet, for which we used our pot-equity.
- In the second article we ran calculations on situations where we bluff, for which fold-equity was required.
So far we have used only one of the two types of equity in our calculations. There are many situations in poker where both types of equity will be needed in order to calculate expected value.
How do we use both fold-equity and pot-equity in the same EV formula?
There is $100 in the pot on the turn. We shove all-in for our last $80. We hold a flush draw and expect to have about 20% pot-equity if we are called. However we expect our opponent to be folding 35% of the time. What is the expected-value of this semi-bluff?
How to break it down
The easiest way to approach problems of this nature (although not necessarily the fastest) is to break them down into separate stages. We can do so using a flow-chart – don't worry if it doesn't make sense just yet.
Using this chart we have shown that there are three possible outcomes to the proposed scenario.
1 – Hero shoves and villain folds.
2 – Hero shoves, villain calls, hero wins.
3 – Hero shoves, villain calls, hero loses.
We can also calculate how often each of these scenarios occur. Hopefully you remember the rule from article 2 for calculating probability of simultaneous events. We simply multiply the respective probabilities in their decimal format.
- So for outcome three to occur, first villain needs to call -which happens 65% of the time - then, villain's hand needs to hold up - which will happen 80% of the time. The probability of outcome 3 happening is therefore →
0.65 * 0.8 = 0.52 or 52%
- The probability of outcome 2 occurring would be -->
0.65 * 0.2 = 0.13 or 13%
- Let's once again remind ourselves of the basic formula for EV. Even advanced calculations stem from the same basic formula →
(P(winning) x #win-amount) – (P(losing) x #lose-amount) = EV
- The difference is very simply that we now have two ways to win instead of one. We can modify the formula by having a separate bracket for each of the three scenarios. Numbers 1,2,3 have been added to the formula to help you see which of the 3 possible outcomes each section of the formula is representing.
(P(winning1) x #win-amount1) + (P(winning2) x #win-amount2) – (P(losing3) x #lose-amount3) = EV
Putting the formula together
So now we have our formula, let's find the values we will need to input. Remember that for the probability of each outcome we need to use our value for the overall chance of that outcome occurring. (We can basically take all the numbers we need from the far right hand side of the flow-chart)
P(winning1) = 0.35
#win-amount1 = $100 (The amount in the pot)
P(winnng2) = 0.13
#win-amount2 = $180 ($100 in the pot + villains $80 call)
P(losing3) = 0.52
#lose-amount3 = $80 (The amount we invested in our semi-bluff)
(0.35 x $100) + (0.13 x $180) – (0.52 x $80)
$35 + $23.4 - $41.6
Adjusting the EV formula
- We don't actually NEED the extra bracket in our formula to calculate the expected value in this scenario – we could just use one bracket for (P(winning) x #win-amount) as we have been previously. The problem arises when we try to find a value for #win-amount. In outcome 1 we win $100, while in outcome 2 we win $180. So which value would we use for #win-amount?
- The answer is neither. One thing we could do is use a “weighted-average” for #win-amount. (If you are unsure how to calculate averages see the section beneath the article entitled “calculating averages”. Knowledge of averages will also be required for article 4 in this series.)
- We win $100 35% of the time and $180 13% of the time. We can't just take the average of $100 and $180 because they occur with different frequencies. We need the weighted average. We have the equivalent to a series of 48 numbers where $100 appears 35 times and $180 appears 13 times.
If we add this series of 48 numbers together and divide by 48 we get:
(35 x $100) + (48 x $13) = 121.66666667
121.666667 would be the average amount of chips we make when we win. We could then use this as our value for win-amount. P(winning) would equal the combined likelihood of outcome 1 OR outcome 2 occurring which can be obtained by adding them together. 35 + 13 = 48%.
NOTE - in probability when we want to find the probability of 1 event occurring AND THEN another event occurring we multiply the probabilities. When we want to know the probability of one event OR another occurring, we add the probabilities.
If we take our new values for P(winning) and #win-amount and put them into the EV formula (this time with only 2 brackets again) --->
(0.48 x $121.6666667) – (0.52 x $80)
$58.4 - $41.6 = $16.8
Hopefully this demonstrates two things -
1) The additional bracket is not required
2) It might be easier to include an extra bracket for every possible outcome. (This is especially useful when we have even more than 3 outcomes) .
Add or Subtract?
If we are going to add a bracket for every possible outcome, how do we know whether to add or subtract them? It's pretty straight-forward. Add brackets which represent winnings, subtract brackets which represent losses.
Another option is to add all the brackets but represent losses as negative numbers. Adding a negative number is equivalent to subtracting it. Note that all the added brackets follow the same format P(winning/losing) x #win/lose-amount)
Let's consider another similar example which you can attempt to work out yourself. The answer is listed below.
1. There is 100bb pot on the turn. Villain shoves all-in for 80bb. Hero check-calls with 20% equity. What is hero's EV?
2. This time hero shoves for 80bb into the 100bb pot. Villain folds 50% of the time, but when villain does call he has 80% equity. What is hero's EV?
Q1 - (0.2 x 180bb) – (0.8 x 80bb) =
36 - 64 = -28bb (hero loses 28bb on average)
Q2 – 3 outcomes. Villain folds 50%. Villain calls and wins 40%. Villain calls and loses 10%.
(0.5 x 100bb) + (0.1 x 180bb) – (0.4 x 80bb)
50 + 18 - 32 = 36bb
We can see the power of fold equity here. Being the one betting turned the situation from a non-profitable one into a profitable one. Aggression is good.
If you were savvy you may have realised it was possible to use your answer from Q1 in calculating Q2. The EV of the all-in scenario was -28bb and this happens 50% of the time. The other 50% of the time you make 100bb when villain folds.
(0.5 x 100bb) + (-28 x 0.5)
50 + -14 = 36bb
- You may find this method easier to use. It effectively uses one bracket for your fold-equity and one for your pot-equity rather than brackets for wins and losses. Remember to weight each value accordingly - if an event, i.e getting to showdown, happens 40% of the time (in total!) you need to multiply the expectation by 0.4.
- If you analyse the EV formula you will see that in reality it can be written as EV = P(winning) x #win-amount. (Where #win-amount takes into account both your wins and losses). The second bracket we've added in our standard EV formula is not technically necessary either, but makes the formula easier to use/understand – just as adding one bracket for every possible outcome may simplify difficult EV calculations.
You may have noticed that thus far, any time we have been calling we have been dealing with an all-in situation. Why is this the case? Because if there are stacks still left in play after we make a call it changes the expectation of our call. We will consider this in article 4.
- There are various different types of “averages” in mathematics – for example “mean, median, mode” to name a few. We are primarily interested in the “mean”.
- The “mean” is calculated by adding all the numbers together in a selection of data and then dividing by the total amount of numbers there are.
- Mean = Total of all numbers ÷ Total amount of numbers
To test your understanding try calculating the “mean” of the following numbers. 2,10,16,21,31
ANSWER – The total of all the numbers is 2+10+16+21+31= 80
There are 5 numbers in total so 80 ÷ 5 = 16
The “mean” or average number of this set of data is 16.