Poker Math: Expected Value with Complete Information

In any game of poker, our main objective is to maximize our profit. For any individual hand we play, we do not know what the outcome will be. However, if we know the probability of winning [P (W)], losing [P (L)], and tying [P (T)] we can determine the average amount we will gain or lose for any decision. The average gain or loss for a decision is its expected value (EV). To be truly successful in poker you must understand EV and use it for all of your poker decisions.

EV is the amount of money that you will win or lose for this wager over the long run. It is important to remember that you are looking at the amount you will win per wager over the long run. For example, if half the time you are going to win $10 on a bet, and half the time you are going to lose $10 on a bet you would expect that over the long term you will break even with this wager. Now, it is possible that you can win 10 times in a row resulting in a $100 profit, but over the long run, you would expect to break even.

To calculate EV you use the following formula:

EV= P (W) + P (L) + P (T)

For example, suppose that with a given wager you will win $100 70% of the time, lose $20 20% of the time and win $50 10% of the time when you tie. Over the long run you would expect to profit $71.00 for each event, as shown by the formula; EV=.70($100) + .20(-$20) + .10($50). Again, it is possible you will win 20 times in a row and profit $2,000, or lose 20 times in a row losing $400, but over the long run you expect to win $71.00 per event.

March 19, 2009- I recieved a comment that my analysis was vague and would be helped with an example. Hopefully this will help:

Look at the following example in No Limit Hold 'em: Let's assume that you have a draw to the nut flush (hearts) on the turn. There is $200 in the pot and you have to call a $40 bet to see the river. You believe that you will only win the pot if you complete your flush. You further believe that if you complete your flush you opponent will fold to any bet and the most you can win is the $200 in the pot.

The first step to determining the expectation from a call is to determine your chances of a win, loss or tie. To win you will need a heart to complete your flush. Since you know what six cards are (your two hole cards, plus the four on the board) there are 46 unknown cards that can appear for the river. Nine of these cards will give you a win (13 hearts in the deck, 4 are already in play). You can determine that you will win roughly 19.6% of the time (9 divided by 46). Following our formula EV= P (W) + P (L) + P (T) you determine the following:

EV = .196 (200) + .804 (-40) + 0 (0) or
EV = $39.13 - $32.17
EV = $6.96

So, in this example you would expect an average profit of $6.96 when you make this call. Remember, it is possible that you could lose 100 times in a row and lose $4,000 or win 100 times in a row and win $20,000, but on average you will win about one out of five for an average of $6.96.

Poker Math- Calling on a Draw

Many times when you're playing poker you find yourself in a position where you know that you can't be winning the hand, but have a chance if the right card comes. Sometimes calling is the right move, sometimes it's not. Here's an example of a hand I played recently:

One of the games I frequent is a $2/$5 no-limit hold'em game with lots of loose action and zero rake. Maximum buy-in is $500 and several players had rebought a few times. With 8 people at the table the average stack was about $750. In this game I try to see a lot of cheap flops with position and play "hit-to-win" poker. I try to win large pots when I hit and run away cheaply when I don't.

I am in the big blind with 6-8 suited (spades), under the gun raises to $10 and gets 4 callers, including the small blind. There is $45 in the pot (4 callers, plus my big blind) so I add my $5 making the pot $50.

The flop comes 2S, 7S, 9H. I've got nothing but I do have an outside straight draw for the nuts and a draw to a medium flush.

The small blind and I check and the original preflop bettor bets $10 into the $50 pot. He gets 4 callers bringing the pot to $90 and I add my $10 to make it an even $100. At this point I have nothing, but 6 cards (non-spade 5 or 10) give me the nuts and 9 cards (any remaining spade) give me a flush. Since I know what 5 cards are (2 in my hand, 3 on the board) there are 47 left to choose from. I hit the nuts 13% of the time and a flush 19% of the time. I figure either of these hands will likely win so about a third of the time I'm good.

The turn comes King of Diamonds. The original bettors eyes light up and I think he has AK or KK. I still have nothing but a draw. The small blind and I check. The preflop bettor decides to slow play and checks, the other two players check. I've gotten a free ride to the river.

The river is a beautiful 5 of hearts giving me the nut straight. The small blind and I check and the preflop bettor throws in $20. He gets one caller and it comes to me with $140 in the pot. I know the preflop bettor has a good hand and will call so I make raise to $160. The preflop raiser smuggly declares "all-in" and adds a little over $500. The other player folds and I call.

My straight beats his set of kings and he tells me how much of an idiot I am for calling his $10 bet on the flop, calls me a donkey and a chaser and spends 10 minutes berating me and whining. I announce that I "felt" I'd get lucky to add to my donkey persona. The extra $760 in front of me helps soothe my hurt feelings.

Now, let's look at my play and decide who the idiot was.

Preflop it cost me $5 to join a $45 pot. Getting 9 to one odds I'd play with nearly any two cards. With suited straight cards I'd probably play even if I knew the other player had aces. It's a very small bet for a potentially big prize.

On the flop 6 of 47 cards gave me the nuts (13%), and 9 cards give me a flush (19%). I know the straight gives me a win and the flush likely gives me a win. 15 of 47 cards are good for me. I'm going to win this pot about half the time if I go to the river, unless someone has a bigger flush draw. I have to play the flush conservatively, but I feel pretty good. when play comes to me I have to pay $10 for a shot at $90. 9 to 1 is a very good payout here.

On the turn there are still 6 of 46 cards that give me the nuts. A little more than 1 out of 8 times I'm a lock. There are 9 cards that give me a flush, but two of those give the preflop bettor a boat or quads. So 7 of 46 give me a better hand than the bettor. Again, a little more than 1 of 8 puts me ahead, I just have to be careful of bigger flushes. I figure I will win 13 or 46 times, or about 3.54 to 1. As long as the pot is 3 1/2 times bigger than the bet I should call. A bet of $50 would give me 3 to 1, causing me to fold, but he bets zero slow playing his set giving me a free look.

The river was fun. All the bettor was looking for was a flush. It didn't hit and ha makes a bet the he figured would get callers giving him a nice pot. When I raised he didn't even think about the possiblity of a straight because there is no way I would have called his bet with a 6-8. My check-raise should have made some bells go off, but there was no way his ego would let him fold a set, or even smooth call. He had to push. Really made me look dumb, didn't he?