Mathematical Expectation
Mathematical expectation is the basis for every decision you will make when
playing poker. Mathematical expectation or expected value (EV), is defined as:
The sum of the value of a random variable with each value multiplied
by its probability of occurrence.
We can express this with the equation:
EV = ((PW) * (AW) - (PL) * (AL)) where
PW = the probability of winning
AW = the amount won
PL = the probability of losing
AL = the amount lost
Let me give you a simple example. A person offers to flip a coin, if it lands on
heads you will owe him $10, and vice versa. Your mathematical expectation
for one coin flip would be the sum of each possible result (+$10 and -$10)
multiplied by the chance of its occurrence, which in the case is 50%. So our
expected value in this instance is:
EV = (0.5 * $10) - (0.5 * $10)
EV = $5.0 - $5.0
EV = $0
So, what we have learned here. The amount we expect to profit by flipping
coins is exactly $0. It makes sense that you could win or loss but in the long
run, you will break even.
Now let's suppose the person offers to pay you $20 if tails comes, but you still
only owe him $10 if it comes up heads. This changes your EV significantly.
The new EV is:
EV = (0.5 * $20) - (0.5 * $10)
EV = $10 - $ 5
EV = $5
Now you know the basic concept about the expectation. Let's apply to poker.
You are holding K♠J♠, and the flop brings a A♠7♠4♣. There is $20 in the pot,
and your opponent bets $20. Do you call? From the previous chapter, we
know that you are about 4:1 against completing the flush on the turn, and
you are getting 2:1 pot odds, according to what we have learned, you should
fold. The decision to fold is based on expectation. A call here, based on pot
odds, has a negative expectation. The odds is 38:9. If you call, you will win
the $40 in the pot nine times and lose the $20 pot it cost you to call the 38
times you didn't make the flush.
EV = (9/47 * $40) - (38/47 * $20)
EV = -$8.51
In this case, you lose $8.51 on average every time you call in this situation.
So, what amount would be acceptable to call?
(9/47 * 40) = 38 * (X/47)
7.66 = 0.81X
X = $9.47
So, if your opponent bet $9.47 or less, you could call with a positive
expectation.
Ok, here are all the basic poker mathematics, now it's time to practice what
we have learned so far before going to the next level. If you really want to
make a living from poker. I suggest you at least practice these basic
mathematics one to two months. Go to PokerStars and open a real money
account and start playing the low fixed limit table and work hard on the
calculation. Play ten people table only because you need more time to do the
math in the beginning. Once you finish the $50 bonus requirement, around
one or two months at $0.10/$0.20 or $0.25/$0.50 fixed limit table, then
come back to read the next level of poker mathematics. You probably could
win some extra by using these math skills because people usually play so
many negative expectation hands in this level.
Click here for more information about PokerStars.
Mathematical expectation is the basis for every decision you will make when
playing poker. Mathematical expectation or expected value (EV), is defined as:
The sum of the value of a random variable with each value multiplied
by its probability of occurrence.
We can express this with the equation:
EV = ((PW) * (AW) - (PL) * (AL)) where
PW = the probability of winning
AW = the amount won
PL = the probability of losing
AL = the amount lost
Let me give you a simple example. A person offers to flip a coin, if it lands on
heads you will owe him $10, and vice versa. Your mathematical expectation
for one coin flip would be the sum of each possible result (+$10 and -$10)
multiplied by the chance of its occurrence, which in the case is 50%. So our
expected value in this instance is:
EV = (0.5 * $10) - (0.5 * $10)
EV = $5.0 - $5.0
EV = $0
So, what we have learned here. The amount we expect to profit by flipping
coins is exactly $0. It makes sense that you could win or loss but in the long
run, you will break even.
Now let's suppose the person offers to pay you $20 if tails comes, but you still
only owe him $10 if it comes up heads. This changes your EV significantly.
The new EV is:
EV = (0.5 * $20) - (0.5 * $10)
EV = $10 - $ 5
EV = $5
Now you know the basic concept about the expectation. Let's apply to poker.
You are holding K♠J♠, and the flop brings a A♠7♠4♣. There is $20 in the pot,
and your opponent bets $20. Do you call? From the previous chapter, we
know that you are about 4:1 against completing the flush on the turn, and
you are getting 2:1 pot odds, according to what we have learned, you should
fold. The decision to fold is based on expectation. A call here, based on pot
odds, has a negative expectation. The odds is 38:9. If you call, you will win
the $40 in the pot nine times and lose the $20 pot it cost you to call the 38
times you didn't make the flush.
EV = (9/47 * $40) - (38/47 * $20)
EV = -$8.51
In this case, you lose $8.51 on average every time you call in this situation.
So, what amount would be acceptable to call?
(9/47 * 40) = 38 * (X/47)
7.66 = 0.81X
X = $9.47
So, if your opponent bet $9.47 or less, you could call with a positive
expectation.
Ok, here are all the basic poker mathematics, now it's time to practice what
we have learned so far before going to the next level. If you really want to
make a living from poker. I suggest you at least practice these basic
mathematics one to two months. Go to PokerStars and open a real money
account and start playing the low fixed limit table and work hard on the
calculation. Play ten people table only because you need more time to do the
math in the beginning. Once you finish the $50 bonus requirement, around
one or two months at $0.10/$0.20 or $0.25/$0.50 fixed limit table, then
come back to read the next level of poker mathematics. You probably could
win some extra by using these math skills because people usually play so
many negative expectation hands in this level.
Click here for more information about PokerStars.
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