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Now I’d like to introduce a term that is the bedrock of profitable
gambling: expectation value. This is commonly called EV for
short. Expectation value is the average amount of money you
can expect to win or lose when you make a wager. If the
average amount is less than zero, then you’re going to lose
money on average. This is called a negative EV bet (-EV). If
the average amount is more than zero, then you’re going to make
money on average. This is called a plus EV bet (+EV). A wager
right at zero is called neutral EV. The goal of profitable
gambling is to be involved in +EV bets as much as possible and
to avoid –EV bets. Your next question should be “How do I
know if the bet is +EV or –EV?” Here’s how. There are three
steps.
1. Identify each possible outcome and the probability of it
happening.
2. Multiply the probability of each outcome by the result it
has.
3. Add together all the results from step two.
Let’s get back to our coin story and go through these three steps
to find the EV of this wager.
1. The coin only has two possible outcomes, heads and
tails. They both will happen 50% of the time. It’s
easiest for me to use the decimal for the probability,
which is 0.5 in this case. Notice the percentages added
together equal 100%. This is very important because
there are no other possible outcomes (assuming it never
lands on its edge). So, step one is complete, and we’re
on to step two.
2. If the coin lands on heads, we will lose $10. We then
multiply this result by the probability of it happening.
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0.5 * (-$10) = (-$5)
Now let’s do the tails. When the coin lands on tails, we
will win $15. This happens 50% as the time as well.
0.5 * $15 = $7.5
Step two is done, and we’re on to step three.
3. Here we add together the results from step two.
(-$5) + $7.5 = $2.5
Our expected value for accepting this bet is $2.50.
Notice that if we just flip the coin one time, we will never have
$2.50. The EV tells us how what we can expect to win on
average if we accept this bet. Since the EV is more than zero,
this bet is called a +EV bet. Being involved in +EV bets is smart
gambling. You want to be able to make +EV bets in poker and
then mass produce them. Imagine you could flip this coin with
this wager five times a minute. That would be $12.50 a minute.
If you could flip it for an hour, that would be $750 an hour.
Mass-producing +EV bets is pretty powerful stuff. Just looking
at the chandeliers when you walk through a casino door will tell
you that. Let’s look at one more example.
A friend gives you a die and offers to pay you $3 any time you
roll a 2. However, if do not roll a 2, you will owe him $1. Let’s
check the EV of this wager.
1. We have one outcome for each number on the die. Each
number has a probability of
1
6
or 0.166 each.
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2. Rolling a 2 has an outcome of winning $3. Rolling a 1,
3, 4, 5 or 6 all have an outcome of losing $1.
Rolling a 1: 0.166 * (-$1) = (-$0.166)
Rolling a 2: 0.166 * $3 = $0.498
Rolling a 3: 0.166 * (-$1) = (-$0.166)
Rolling a 4: 0.166 * (-$1) = (-$0.166)
Rolling a 5: 0.166 * (-$1) = (-$0.166)
Rolling a 6: 0.166 * (-$1) = (-$0.166)
3. Let’s add together the results from step two.
(-$0.166)+ $0.498 + (-$0.166) + (-$0.166) + (-$0.166) +
(-$0.166) = (-$0.332)
We come up with about negative 33 cents a roll. The EV is
below zero, so it’s a -EV bet. These are not the types of bets we
want to take. We can also see the power of mass-producing -EV
bets. If we could roll this five times a minute, we would lose
$1.65 every minute. Rolling at this pace for an hour would give
us a rate of losing $99 an hour. This doesn’t sound like the type
of bet I want to take!
Making money in poker is all about being involved in +EV plays
as often as possible and avoiding -EV plays as often as possible.
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Quiz
(Answers on pg. 158)
1. Someone has the four As face down on a table. You
have one chanc
e to try to pick the A♠
. If you pick it
correctly, they’ll pay you $3. If you do not, you pay $1.
What is the EV of this wager?
2. There are three cups upside down on a table.
Underneath one is a green ball. Underneath another is a
red ball. Underneath another is an orange ball. If you
pick green, you win $5. Pick red, you lose $2. Pick
orange, you lose $1. What is the EV of picking one cup?
3. Someone holds out a deck of cards. If you pick out a K,
they’ll give you $10. If you do not, you owe them $1.
What is the EV of this wager?
4. Someone holds out a deck of cards. If you pick out a
spade, they will give you $4. However, if you pick out
the A♠
, they’ll give you $20. It will cost you $1 to draw.
What is the EV of this wager?
5. Someone gives you two dice. They offer to pay you $37
if you roll two 6s. However, it will cost you $1 a roll.
What is the EV of this wager?
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Hit the Deck
Counting Outs
We’ve now laid the foundation to begin understanding how to
make good decisions when having the opportunity to make a
wager. Now it’s time to begin applying this math to poker.
When watching poker on TV, you’ll often see a percentage next
to the players’ cards. The percentage is letting the viewers know
how often each player is going to win the hand by the river. I’ve
often heard new players say “If I could only know those
percentages, I could do alright.” Well, there’s a lot more to
playing good poker than knowing what those percentages are;
however, it is a critical skill to be able to estimate that
percentage fairly accurately. The start of this process is by being
able to count outs.
What is an out? An out is a card that can come on a future
street(s) that can give you the best hand. So, thinking about outs
only applies when you do not have the best hand and there are
more cards to be dealt. Let’s say you are playing a hand, and
you’re on the turn.
Hero: Q♠T♥
Villain: K
♦
7
♣
Board: 6
♠
9
♦
2
♣
K
♥
You’re the hero, and your opponent is the villain. It just makes
sense, we’re the good guys, and they’re the bad guys, right? The
villain has a pair of Ks, and you only have Q high. So you
definitely do not have the best hand. However, there is one card
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36
left to come on the river. What cards will give you the best
hand? Neither a Q nor a T will help you because neither pair
will beat his pair of Ks. However, if a J falls on the river, that
will give you a straight for the best hand. So, a J is an out for
you. How many Js are left in the deck? There are four of them.
So, you have four outs in this hand. Being able to count or
estimate your outs is a critical skill in poker. Let’s count our
outs in this hand.
Hero: 5
♠
6
♠
Villain: A
♦
K
♦
Board: A
♠
7
♠
Q
♦
2
♥
How many outs do we have? Neither a 5 nor a 6 on the river
will give you a winning hand, and there’s no way to get a
straight. However, if another spade comes, you will have a
flush. There are 13 spades in a deck, and we see 4 on the board.
This leaves nine other spades in the deck. You have nine outs.
This is pretty simple, but there are several aspects of outs that
people miss. There are backdoor outs, hidden outs and
chopping outs.
Let’s talk a bit about backdoor outs. Backdoor outs apply on the
flop only. They add a little extra value to your hand. But, they
require a combination of turn and river cards that both help your
hand. A classic example is called a backdoor flush draw. Let’s
look at this example.
Hero: 4♠
5
♠
Villain: A
♣
K
♦
Board: K
♥
9
♦
4
♣
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You end up getting all-in on the flop. If we look at your outs
here, you need to get either a 4 or 5 on the turn or river to make
two pair and beat his pair of Ks. There are three 5s left, and two
4s left. This gives you five outs. If you were playing on TV,
they would put 18% next to your hand. Your opponent would
have the remaining 82%. I’ll show you how to get these
percentages in the next section; however, for now, I just want to
show you the impact of a backdoor flush draw. Now, instead of
the board being K94 with the 9
♦
, let’s change it to K94 with the
9
♠
. This is the same suit as our two cards.
Hero: 4♠5♠
Villain: A
♣
K
♦
Board: K
♥9♠4♣
Now, if the turn and river are both spades, you would have a
flush. This is a chance for improvement you didn’t have when
there was no spade on the flop. If this were on TV, you would
have 22% and your opponent would have 78%. This is about a
4% increase for you.
Another backdoor draw is a backdoor straight draw. Let's look
at this example.
Hero: 6♠7♠
Villain: A
♣
K
♦
Board: K
♥
2
♦
6
♣
Again, if you were to get all-in here, you need a 6 or 7 either on
the turn or river. This is a total of five outs, and you again would
have 18% next to your hand, and your opponent would have
82%. However, let's change the board (changing the 2
♦
to a 5
♦
).
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38
Hero: 6♠7♠
Villain: A
♣
K
♦
Board: K
♥
5
♦
6
♣
Now you have the chance for the turn and river to come an 8 and
9, a 4 and 8, or a 3 and 4. Any of those three combinations of
turn and river cards would give you a straight. This is a chance
for improvement you didn’t have before. Now your hand would
have a 23% chance to win, and your opponent would have the
remaining 77%. This is about a 5% improvement from the
previous board.
Some backdoor straight draws are stronger than others. For
example, if the board were K
♥
4
♦
6
♣
, you would still have a
backdoor straight draw. However, your percentage would go
down to 21%. This is because we lost a straight chance. When
the board contained a 5, you had three chances for the straight.
Now with the board having a 4 instead of a 5, you only have two
chances for the straight. The turn and river need to come down
either a 5 and 8 or a 3 and 8. So, this hurts your winning
percentage a bit. The same is true if the board were K
♥
3
♦
6
♣
.
You have the chance for a backdoor straight, but the board must
come specifically a 4 and 5, giving you only one chance for a
straight. With that, your percentage would be 20%. But, these
specifics aren’t nearly as important as recognizing a backdoor
draw and realizing on average it adds about 4%.
Having multiple backdoor draws can add quite a bit of value to a
hand. Let's look at this example.
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39
Hero: 6♠7♠
Villain: A
♣
K
♦
Board: K
♥5♠6♣
Now you have both the backdoor flush draw and the backdoor
straight draw. Now your percentage would change to 26%. This
is about an 8% increase from when you have no backdoor draws.
This is a significant increase and can really impact how you play
a hand. Now, in terms of counting outs, if you’re going to see
both the turn and river, you can add an extra out for a backdoor
draw. So again, looking at your board here, you have the five
outs for the 4s and 5s in the deck, but you also have the backdoor
flush draw and backdoor straight draw. Since you’re all-in and
will be seeing both the turn and the river, you can add one out
for each backdoor draw. So, we can say you have seven outs in
this hand. This is easy enough, but sometimes things are not so
obvious. Sometimes the outs get a little sneaky.
Let’s talk about those hidden outs. Sometimes we have more
outs than we first think. Future cards can often take our
opponent’s cards out of play and vice versa.
Hero: A♠Q♣
Villain: 3
♣
3
♦
Board: T
♥
T
♦8♠5♣
Of course, your opponent has a pair of 3s, and you only have A
high. How many outs do you have? Many times people only
think about an A or Q to give them a pair. There are three As
left, and three Qs left. That would be six outs. However, your
hand is much stronger than that verses 33. If the river came an 8,
you now would win the hand. Your best five cards would be
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