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Expectation
by: Ed Hill

You are playing online poker and the site is running a promotion where all the players that are dealt in on a particular hand will receive a $5,000 bonus and the actual winner of the hand will receive $50,000, can it ever be right to fold?

Well it is definitely not ever correct to fold before the flop, yet you see people do it all the time. What is going on here? Well, these people are novice gamblers that don’t have any conception of the meaning of expectation.

Expectation tells us what our $1 is worth right now at this exact moment. You do not need to be a favorite to win your bet to have a good bet. What you need is a positive expectation. In other words, your $1 needs to be worth more than $1, the odds against you actually winning do not matter!

Say, for example, that you are playing no-limit and one player moves all-in for $100 and for some reason his hand is exposed and it is AA, everybody folds and you hold AK. Now you have one of the worst hands you could possibly have, you would much prefer to 5§6§. So do you want to play the AK?

AK is a 19-1 underdog to win the pot that means that you only have one chance in 20 of winning. 5% is not very good odds, what should we do?

Well, we get $50,000 if we win and lose $100 if we don’t, what is our $100 worth right now at this moment?

The chances of us winning the $50,000 are 5% and the chances of losing the $100 are 95%.
(.05) (50,000) + (.95) (-100) = 2500 –95 = 2405. So, in this case we should play AK when we are looking at AA laying face up in front of us! Even though we are going to lose $100 95 times out of 100 we are getting to make a great bet.

This is the reason why casinos keep getting bigger and bigger, because they are getting to book bets that have a positive expectation.

Universal expectation equation. (Probability) (Return) -$1

Probability is the chance that the event is successful.

Return is how much we are going to get back when the event is successful. When calculating your return always pretend that you are betting exactly $1 and that you have posted. What I mean by posted is that you are have already put your $1 up and are going to get it back when you win.

We have always heard that the house edge in Roulette is 5.26% let’s put it in the expectation equation and see if we come up with that exact answer.

There are 38 spaces on the wheel, the numbers 1-36, plus 0 and 00. If we bet on lucky number 7 we have 1 chance in 38 of winning.

Therefore our probability is 1/38.

We are receiving odds of 35 –1 and have already put our $1 up. Therefore we are going to get $36 if the 7 hits.

(1/38) (36) – 1 = -.0526

What happens if we choose to bet on red or black? Or odd or even?

There are still 38 spaces, 18 are red, 18 are black, and the 2 zeroes are green. There are 18 odd, 18 even, and the 2 zeroes are neither odd or even.

So we bet our $1 on red, what happens?

Our probability is 18/38.

These types of bets pay even money, since we already posted we are going to get back $2 if we win.
(18/38) (2) –1 = -.0526

Now we go over to a friend’s house to play poker, at the end of the night he says, “I will spread the deck face down on the table and give you odds of 60-1 that you can’t pick the Ace of Spades.” Wow, great bet, yes I want to play. Then he says, “I am not going to do this all night, I only want to do it one time.”

So we ask him, “How much can I bet?” His reply, “I don’t care.” So how much do we want to bet on a proposition that we are going to lose 51 times out of 52? We know we have a great bet, the odds are 51-1 against hitting but he is willing to give us odds of 60-1. So we are faced with the proposition of getting to make a great bet that we are probably going to lose. What is the correct bet?

Oh yes, the expectation equation.

The probability is 1/52.

The return is 61. Remember always pretend you posted.

(1/52) (61) –1 = .173

We have a 17.3% edge on this bet but are certainly not willing to bet 17.3% of our bankroll on something that is going to lose 51 times out of 52. Especially since we only get to play one time.
To ascertain the correct mathematical bet simply take the expectation equation and divide that answer by the odds against winning. In this case 51.

.173/51 = .023 Therefore, we can still bet 2.3% of our bankroll on this bet.
Yes, we are probably going to lose this one and sometime in the future we are going to be confronted with other bets that are similar to this. Just keep betting intelligently, making sure you always have the odds in your favor and given enough time you will be a winner.

Are you interested in the math of poker? Read this interesting post about the Fundamental Theorem of Poker

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