The concept of expected value is a very important one in poker and it is a concept that any novice or intermediate player needs to get their head around for one very important reason. This is because you need to assess whether or not you are merely making good decisions or whether you are simply running good or bad. We can calculate expected value in the plays that we make and then compare our wins and losses with those results. So EV is very useful to understand because it allows us to measure far more accurately how well we are playing because actual short term results are often a very poor indicator of skill.
So what is expected value?
Expected value or EV is the sum total of all of the outcomes that are inherent within a particular situation where each individual outcome has a value. Let us look at a simple example here to show you what we are talking about. Imagine if we placed ten balls into a bag and six of those balls were white while four of them were black. If we then said that I had to pay you $100 if you pulled a white ball out of the bag while you had to pay me $100 if you pulled a black ball then what would the expected value (EV) be?
Well firstly you have a 60% chance of drawing a white ball and only 40% chance of drawing a black ball. So if we take an average set of outcomes over ten draws then you would pick a white ball six times out of ten and win $600 or 6 x $100. You would draw the black ball four times out of ten and lose $400 or 4 x $100. So your wins are $600 while your losses are $400. This leaves you with a projected profit of $200 which we then divide by the number of times that you draw a ball out of the bag which is ten. This leaves us with an average profit per draw of $20 and so your EV per draw is $20. You can win or lose $100 over any one draw but your EV is +$20 and this is the long term expected value if we were to replicate this process almost infinitely.
It is often difficult for players to see concepts like these and picture them within a poker setting and so we will look at a hand example here to help us out. Let us say that you have the Ad-9d and the board is Qd-8c-4d-2s giving you the nut flush draw. Firstly we know what our hand is and we also know that we have nine flush cards to make our nut flush. Now we have to look at what is in the pot and let us say that the pot is $70. You are heads up and your opponent shoves all in for $30 making the pot $100. From your knowledge of your opponent then you think that he would only allow himself to get all in like this with made hands that are better than a single pair and would not bluff.
So you deduce that your ace overcard is dead and that you have to make your flush to win the pot. So your options are simple in this example and are either calling or folding. Working out the EV of folding is easy because you are placing no more money into the pot and so you can neither win nor lose on that round of betting and so your EV is 0. But calling is more complex and you have nine flush cards left in the deck with a total of 46 unseen cards left to be dealt. You can use poker software like PokerStove to enter these calculations if you like but the chances of you seeing a diamond on the river to make your flush are 19.6% which also means that you are around 80% not to make your flush. We can round these numbers off for simplicity to see that our odds of making the hand are around 4-1.
Using approximations in poker is very good because in actual play then this is what you will be doing anyway. So you will make your hand here 20% of the time which means that your equity is 20% of what is in the pot at this time which is now $100. So you make $20 from this situation but it will cost you $30 to make the call. So we can now deduce the EV of this play and you are looking at calling the $30 bet in a pot with $100 in it with a hand that you will make only 20% of the time and making $20 from it.
So you are losing $10 in value by calling and so your EV for calling is -$10 which is clearly a worse result than the EV for folding which as you recall was $0 and so the best play here is to fold and this is the concept of expected value.
Written by: Carl “The Dean” Sampson – Poker author, writer and online poker pro