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What is an Expected Value?

Poker Expected Value

Expected value is the theoretical worth of an event that is repeated an infinite number of times, and it’s the foundation of gambling.

The Law of Large Numbers says that if an event is repeated many times the outcome of that event will be about the same as the theoretical probability of that outcome. This allows us to use expected value to evaluate specific options in the play of poker hands, and use it to determine the most profitable option.

Many people think of the Law of Large Numbers as the “law of averages.” Although the “law of averages” has become widely accepted outside of academia and the serious study of probability, within academia and the serious study of probability it does not exist.

In order to calculate expected value we must be able to establish our sample space, and we must be able to break that sample space down into its relevant events. Once we’ve identified each event, we determine whether it’s a positive event, or a negative event.

Then we establish the value of each event. With that complete, we just simply add together all of the values for every event. If we get a positive number we’re earning money on that bet. If we get a negative value we’re losing money. By dividing the total amount won or lost by the sample space, we get our expected value, or expectation, for each event. That’s the amount we can expect to earn every time we make that particular bet, regardless of whether or not we collect the chips.

This will always be the way to calculate the expected value. In poker, as with all other games of chance, we want to have a positive expectation as often as possible.

Flush Example

Here’s an example to introduce you to expected value calculations. The situation is deliberately simplified to make the calculations simple, in order to make the concept easy to understand.

Here’s the situation. You have a four-flush on 4th street against one opponent. You each put in one bet before the flop, on the flop, and on 4th street. If you make the flush your opponent will call your 5th street bet, and you will win. If you do not make the flush you will fold. You and your opponent are in the blinds, that way there is no other money in the pot to consider. The question is, what is the value of your hand under these circumstances?

For this example, your opponent’s cards are not defined, so the sample space is 46. If your opponent’s cards were defined, the sample space would be 44. Of the 46 remaining cards, 9 make your flush, and 37 don’t.

Let’s use 2-4 limit for this example. The 37 times you miss the flush and lose the pot, you lose 8 dollars. That’s 2 dollars before the flop, 2 dollars on the flop, and 4 dollars on 4th street. For a total of (37 x 8 = 296) 296 dollars lost.
The nine-times you make the flush and win the pot, you win 12 dollars. That’s 2 dollars before the flop, 2 dollars on the flop, 4 dollars on 4th street, and 4 dollars on 5th street. For a total of (9 x 12 = 108) 108 dollars won.

Your expected value is the sum of your 296 dollars lost and your 108 dollars won (296 – 108 = 188), which is a loss of 188 dollars. The 188-dollar loss divided by the sample space of 46 reveals that on average you would lose about 4 dollars per hand.

Steven James is the author of The Evolution of a Poker Player.