Expected Value

Expected Value, often written as EV, is a probability term used by poker players that is often the core consideration when choosing to call, fold, or raise. The expected value of a choice is the weighted average of all the possible outcomes. It is usually expressed as, for example, +\$350 EV or -\$100 EV, but is often shortened by poker players to just +EV (positive expectation) or -EV (negative expectation) since the decision to call or fold is typically based on whether you win or lose money, not by how much you win or lose.

It's calculated by adding up the sums of each outcome, multiplied by the chance of that outcome occurring. It's also easier to describe with examples.

Let's say we're playing a game where you draw from a standard deck of 52 cards. If you draw an ace, king, queen or jack, I'll pay you ten dollars. If you draw a numbered card, you pay me five dollars. Hopefully you can figure out that the chance of drawing any numbered card is 36/52 and the chance of drawing a jack through ace is 16/52. To get the expected value of a draw, you multiply those probabilities with their actual payouts...

• Jacks through Aces - (16/52)*\$10 = +\$160/52
• Numbered Cards - (36/52)*-\$5 = -\$180/52

...then add them together. It comes out to (-\$20/52) or ~ -\$0.38. That means that, on average, you'll lose 38 cents per draw. You obviously can't lose 38 cents though. You'll either lose \$5 or win \$10. This just states the exact average loss. If you played the game a thousand times, you would very likely be close to that 38 cent per draw number. If you played with exactly one deck without redrawing, you would be precisely at that figure. More importantly though, in simply knowing that the game has a negative expectation, you know not to play no matter how much the losses are.

Texas Holdem Example

We run into scenarios in holdem frequently where we can calculate our EV. This is a great exercise to do after the hand is over that helps us quantify those close decisions we run into. Here's a good example.

We have... ...with a board of...

The pot is \$100 and your sole opponent's all-in bet is \$100, which is less than your stack. In the worst-case scenario, he has a set. Assuming that's where we are, of the 990 possible turn/river combinations, we win 335 with our gutshot and flush draws. Our opponent wins 655 of them. (Sample) If we call and win, we win an extra \$200. If we call and lose, we lose \$100. What would our estimated value be in that case?

• Win - (335/990)*\$200 = \$67000/990
• Lose - (655/990)*-\$100 = -\$65500/900

So our EV ended up being +\$1500/990 or +\$1.52 on average every time you call. Compared to a \$100 bet, \$1.52 is fairly insignificant. However the fact that it has any positive expectation means that it is correct to call here from a strictly mathematical sense. It is, after all, higher than zero, which is our EV if we fold. It also doesn't hurt to know that we're +EV against his best possible cards. With any other cards that \$1.52 goes way up.

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