Expected value

What this fancy looking expression mean is how much money you will make or lose in the long run on average every time you take a specific action. Let's forget about poker for a second and suppose I suggest you to throw a dice 100 times, every time you get 1 or 2, you give me 10$, every time you get 3, 4, 5 or 6, you give me 8$, what should you do? Of course you should refuse, on average, you will lose:

(33.3% of time * win 10$) - (66.6% of the time * lose 8$) = 1.80$ lost every time you throw that dice!

That is a negative expected value of 1.80$. Now lets go back to poker. If there is only 1 sentence that you should remember of this whole website, it should be this one:

The best decision that you take is simply the one that has the highest expected value.

There is no better reason than this to play a hand a certain way! I don't care if you say you wanted to make sure he does not get his straight, if playing some other way would have a higher expected value, then playing some other way is a better decision that what you just did, period! There is no better way to play a hand than the way that has the highest expected value and any other way of playing the hand is inferior!

"But see, I told you, if I would have played your way, I would have lost 10$ more!" Well my friend, I would have lost 10$ more this time, but those 5 other times I was in a similar situation, I make 5$ more than you every time! Getting the point?

There is no better way to learn about expected value than doing some math's with real examples!

You are in the "advanced stuff" section of my site, so I don't have to "go easy" on you, let's start by a situation I like:

1$-2$ no limit game, your opponent limps, you know he does that with weak Aces, suited connectors and low pairs. You raise to 8$ with Ad Qd, he calls and everybody else folded, the flop is 2d 8d Tc giving flush a flush draw with 2 overcards. Your opponents bets 10$, you are now pretty sure he has top pair, what do you do now? Well, of course you think about expected value, which action has the best expected value, is it fold, call or raise?

Fold: well, it's easy, it's 0$ expected value!

Call: The guy most probably has AT, JTs or T9s which gives you a probability of winning in the 53% range. We could start by saying that you will win that 29$ on the table 53% of the time and lose another 10$ 47% of the time, which gives an expected value of +10.67$. In fact, it is a bit worse for a few reasons, first one being if you don't hit the turn, he will charge you again for the river. Second, if the flush card hits or a cards over T, you will have a have time getting paid some more. Lets make an approximation, 8$ expected value.

Raise to 25$: Reraise with a flush draw, are you crazy??? No, I'm not, I don't care if it's reraising with a draw as long as it has the best expected value, so let's find out if it really has! The guy has flopped top pair, T with most probably a 9, J or A kicker, he bets 10$ and you reraised him to 25$. You raised preflop, he will fear a better kicker if he does not have the Ace and he will definitely fear an overpair too. Lets say he will fold 50% of the time and call the other 50%. Now, you win the current 29$ pot 50% of the time, off the remaining 50% of the time, you win the 29$ pot 53% of the time and lose 10$ more 47% of the time, which gives us: (50% * 29$) + (50% * 53% * 44$) - (50% * 47% * 25$) = 20.29$. Now, that's a lot better than 8$ or 0$ don't you think? And also, think about how easier it will be to get paid if he does call and you get your flush! Want another good news, you're setting up for a future hand! When your opponent will notice that you just reraised him with a flush draw, you're giving him good reasons to give you his whole stack on top pair, remember that when your hit your next set!

Now that's how a shark thinks! In this last example, the flush draw was nice, but the 2 overcards giving us 6 additional outs sure helped! Lets try this again without the 2 overcards, same hand but you have 3d 4d (remember: flop 2d 8d Tc, your opponent has top pair).

Fold: Still 0$

Call: (Win 38% * 29$) - (Lose 62% * 10$) = +4.82$

Raise to 25$: (Win 50% * 29$) + (Win 50% * 38% * 44$) - (Lose 50% * 62% * 25$) = +15.11$

Wow! Still without the 2 overcards, it brings much more money to reraise with the statistically inferior hand than to fold or call. Starting to get a feel what this "aggression" is all about? Now do remember these calculations were based on the fact that there is 50% chances that the other guy will fold to the reraise, don't do this against a player that almost never folds!

Also, remember this article about helping your opponent's make mistakes. Well here's another good part about raising that flush draw... when he calls, most of the time (about 80% of the time), you will not hit your flush on the turn. Now, since you only have 20% chances of hitting it on the river, your opponent's best move would be to make a big bet, but... if he hits hard, he will want to check raise you since he thinks you are likely to bet (you reraised him on the flop). If he does not improve his hand, why would he bet against you after you showed so much strength on the flop? He will most probably and give you the opportunity to have a free river (or to bet again if you have reasons to think he might fold).

Enough math's for now, you understand what is the expected value and you know that every decision you make has to make this as high as possible, that's good enough for now! If you feel like you just learned a new aspect of the game I suggest you get some practice, back-up this new knowledge with some real game experience before you continue on reading.