You'll need to understand expected value as a concept so pay attention closely (also referred to as "expectation", don't get confused). This is basically how you decide whether there was/is positive value in your play. And that's a big deal.
As poker players, we're always in the look for more value in the way we play our hands. By calculating expected value, we can easily play around with different kind of playing strategies and this way come up with the most profitable, the most valuable ways to play poker. We're always trying to get as much expected value as possible. The whole game of poker is about getting as much expected value as possible, that's ultimately what our strategies, tactics and theories are for.
I'm going to start with a simple coin flip example. I think you'll get the idea of expected value just from this example, but we'll go through some more just to beat the point home. Anyway, let's say you and your friend flip a coin, and you'll get $75 everytime you win, he gets $25 everytime he wins. Obviously you're going to be a winner in this situation, but how much is each coin flip worth to you? Since the chances for you to win are 50%, and chances for your friend to win are 50%, too, you'll need to use following simple math:
$75 * 0.5 = $37.5
$25 * 0.5 = $12.5
$37.5 - $12.5 = $25
This means your expected value per every coinflip is $25.
Expected Value in Poker
In order to be successful at calculating expected value in poker, you'll have to know a thing or two about poker hand ranges. Why? Because otherwise there's no way of knowing how many times of how many times we're going to have a winning hand. Figuring out hand ranges has a lot to do with poker mathematics, too, but being good at the psychological side of poker helps you a lot with it.
An Example
For sake of making this article easier to understand, we'll set very strict hand ranges for our opponents. Let's say he always folds when he's got nothing.
Our hand is 87 of diamonds. The board is 3s 4h 9d Ts Jd.
We've got a straight, like you can see. Right now we don't care about the times our opponent is going to fold, because we want to know what's the expected value for either a bet or a check.
The hands our opponent can have that he won't fold with:
- KQ of spades (KQs) - 1 combination.
- QJ of spades (QJs) - 1 combination.
- AJ - 9 combinations.
- AT - 9 combinations.
Let's count the possible combinations for these hands: 1 combination of KQs, 1 combination of QJs, 9 combinations of AJ and 9 combinations of AT.
Then we add them together: 1 + 1 + 9 + 9 = 20.
Now, the pot is $150 and we act first. If we bet, we either get called or raised. KQs is obviously going to raise (in which case we're going to call). To make this real simple, AJ, QJ and AT will always call.
So, our bet is $110 and there's a 1/20 chance we get raised (and lose $200 more). 1/20*$310 = $15.5.
There's a 19/20 chance we get called and make $110, so 19/20*$110 = $104.5.
Our expected value from betting is $104.5-$15.5 = $89. In other words, our average profit for every bet in this situation is $89.
IF we decided to check here, he'd bet KQs (obviously), AJ, QJ and AT. What if we decided to raise every hand he bets here (for $200? He would call our raise with AJ and re-raise with KQs (here we fold). So:
10 times out of 20 (1/2 or 50%) he bets, we raise and he folds. We make $110. 9 times out of 20 we first check and then raise him, he calls for $200 more. We make $310. Once we raise and he re-raises, then we fold and lose $310.
10/20*$110 + 9/20*$310 - 1/20*310 = $55 + $139.5 - $15.5 = $179. This means our check-raise compared to bet is $74.5 more profitable per time. Our expected value is that much bigger!
Poker Mathematics:
Poker Probability,
Pot Odds
(Continue: Implied Odds),
Expected Value,
Winnings in the Long Run |
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