Expected Value ( Extremely Simple Math)

Expected value is a way of calculating how much we stand to make in a particular scenario. The expected value will be based on our current equity and pot odds when we face a bet. If we are the player making the bet, the expected value will be based our pot equity, bet size and fold equity.

Expected value real life example

To take a real world example – we are parking our car in a city and unfortunately have to pay. The cost is $5 for 2 hours and we know we will be less than two hours. We believe that the probability that we will be caught without a ticket is 10%. The fine for not having a parking ticket is $60.

Should we buy the ticket beforehand? Or take the chance that we will not be caught?

To find the expected value out we will use a simple equation. For people who aren’t too keen on math, don’t worry, it’s pretty simple.

To work out our expected value we multiplying the potential gain by the probability of incurring that gain (for example saving $5 by not paying the parking ticket by 90%) and then multiply the potential loss the probability that the loss will occur. The wording may be quite confusing by the equation will make more sense:

EV = (probability of gain)*(value of gain) + (probability of loss)*(value of loss)

For our parking ticket example this becomes:

EV = (0.90)*($5) + (0.10)*(-$60) = $4.5 – $6 = -$1.5

So on average, every time we don’t pay our parking ticket we will stand to lose $1.5. This may confuse people as in no single case can we lose $1.5 – we either save $5 or we have to pay $60.

The $1.5 comes from is the average loss we will make over a long period of time. So if we did this 100 days in a row, 90 of the days we would have saved $5 each day for $450 , but we would have been fined 10 times for $600 total. Thus on average we would have lost $150 which is $1.5 per day.

Change variables and finding the breakeven point

Using this simple calculation we can see how changing the variables affects our expected value; if it is less likely we will be caught and fined we should not pay for the ticket, that is obvious. For example, it is 1% likely that we will be caught without a ticket:

EV = (0.99)*($5) + (0.01)*(-$60) = $4.95 – $0.6 = $4.35. Therefore it make sense to not buy a ticket for these parameters.

Intuitively that makes sense too since it is so unlikely we will have to pay $60 dollars.

We can also find the point at which it becomes profitable to start not paying for tickets. We find the breakeven point by setting the EV to zero and then finding the probability that we will be caught – x. This is called the breakeven point (requires some algebra):

EV = 0 = (x)*($5) + (1-x)*(-$60) = 5x+60x -60

Therefore: 65x =60 and x = 60/65 = 92.3%.

So if we are likely to be caught greater 7.7% of the time we should buy a ticket; and if less than that we should not buy a ticket. Simple.

We can apply this analysis to poker in situations where we know our equity pot odds, bet size and pot size.

Expected value in poker

To go back to the previous hand example with A9 of diamonds we had pot odds of 28% and we had pot equity of 18%. The pot odds is based on how much he bet and pot equity is based on the hands we assumed he would be bluffing with. So let’s put that all together to do an expected value calculation.

Expected value poker

Again the expected value will be:

EV = (probability of gain)*(value of gain) + (probability of loss)*(value of loss)

The probability of gain will be our equity which is 18%.

The probability of loss will be 1-equity which is 82%.

The value of our gain will be $67.5 and the value of our loss is -$26. Subbing in:

EV = (0.18)*(67.5) + (0.82)*(-$26) = 12.15 – 21.32 = -$9.17

Therefore this will be an unprofitable call to make. That should also be intuitive since we are winning so infrequently.

A fast trick to determine profitability

A quick way to determine profitability of a call without doing a full EV calculation is to compare the pot odds and our equity. In the case of the example hand we have pot odds of 28% and we have equity of 18%. If the equity is less than the pot odds we should not call as it would be unprofitable; conversely if we had more than 28% equity we can make the call as it would be profitable to do so.

You are not expected to do these calculations in their head while at the poker table. The purpose of the calculations is to analyse difficult hands off the table after your poker session, not during a game. This allows you to determine if you made the right decision and correct mistakes.

One of the main drawbacks of EV calculations is that you cannot perform them before the river as there are so many variables at play.  We do not know what our opponent will do on future streets, will he check or will he bet? What card will come on the turn and river? If he does bet, what size will he use?

This is what make poker such a complex game and a difficult one to solve computationally.

Despite this fact, expected value, pot odds and equity is useful in both poker and real life. So it is worth taking the time to understand how it is applied.

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