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Probability Theory
Probability and the ability
to understand and predict human behavior are the two important
keys to playing and winning at Texas Holdem. Learning how
to read your opponents comes with practice, a keen sense of
observation, and years of experience. The knowledge of Texas
Holdem probability theory can and should be learned before
you sit down with a pile of chips in front of you. For example
the odds of catching your flush or straight, the odds of getting
an overcard, or the proportion (or percentage) of times you're
going to match a card on the flop to your pair of cards in
your hand, and the percentage of times you can expect to lose
if you do not catch your set on the flop holding a small pair
are a few extremely important factors in learning how to play
Texas Holdem Poker. Knowledge of these statistics is probably
the most important key to winning and is often the difference
between winning and losing. In online games especially without
any face to face confrontations, statistical knowledge becomes
the main factor when choosing whether to bet, call, or fold.
Here are some terms that you'll
hear on this site and whenever you're talking about poker
odds...
Outs
The number of cards left in
the deck that will improve your hand. "I had four diamonds
on the turn, so I had only 9 outs left to finish that flush."
Pot Odds
The odds you get when analyzing
the current size of the pot vs. your next call. "There's
$240 already in the pot, and another $12 bet coming at me,
so my pot odds are good if I hit that diamond flush"
Bet Odds
The odds you get as a result
of evaluating the number of callers to a raise. "With
a 1 in 5 chance of hitting it, and knowing all six of these
guys are planning on calling my bet, my bet odds are good
too."
Implied Odds
The odds you are getting after
the assumed result of betting for the remainder of the hand.
"Since I think these guys are going to call on the turn
and river, my implied odds are excellent."
In Texas Hold 'Em, you commonly
use outs and pot odds the most. This is also the starting
point for those who want to learn about poker odds. To those
out there who "are not exactly great mathematicians",
you better get good because that is how it's done. At this
point it's only simple division The numerator will be the
number of outs you have. The denominator is the number of
cards left that we haven't seen. The result will be the percentage
chance of making one of those outs. Therefore, the most math
you'll be doing will be dividing small numbers by 50 (pre-flop),
47 (after the flop), or 46 (after the turn).
Before we move on, we would
like to take the opportunity to clarify one point of statistical
interest. A lot of you might wonder why we never factor the
opponents' cards or the nonused cards (referred to as the
burn cards) when figuring out the denominator in our mathematical
interpretation. The answer is very simple. We only consider
"unseen cards" in all our mathematical calculations.
If you saw what the burn cards (or unused cards) were, or
an opponent showed you his hand, you would know that those
cards are not going to be drawn and could use that information
in your calculations. We typically do not know what they have,
so we don't even think about it when talking about odds.
Pot odds are as easy as computing
outs. You compare your outs or your chance of winning to the
size of the pot. If your chance of winning is significantly
better than the ratio of the pot size to a bet, then you have
good pot odds. If it's lower, then you have bad pot odds.
For example, say you are in a $5/$10 holdem game with Jack-Ten
facing one opponent on the turn. You have an outside straight
draw with a board of 2-5-9-Q, and only the river card left
to make it. Any 8 or any King will finish this straight for
you, so you have 8 outs (four 8's and 4 K's left in the deck)
and 46 unseen cards left. 8/46 is almost the same as a 1 in
6 chance of making it. Your sole opponent bets $10. Now you
have to decide if calling that bet is a good idea or is folding
the move to make here. How do we know what to do? Simple –
look for the math to help you make the right decision. Now
if you take a $10 bet you could win $200. $200/$10 is 20,
so you stand to make 20x more if you call. 1/6 is a greater
number than 1/20 so the math is telling you to call this bet
into you. What about raising into this same scenario? Well
then you would be putting in $20 to make back $220 or 1/11
so again the real gambler would raise since he could either
win with his straight or win when his opponent folds. However
if his opponent re-raised another $60 then you would be getting
only 1 in 4.67 to one odds and that is not enough money to
call – so now it would be time to fold.
Another clarification is in
order right about now ... you see there are a lot of players
want to somehow factor in money they wagered on previous rounds
in all their calculations. With the last example, you probably
had already invested a significant portion of that $200 pot.
Let's say $50. Does that mean you should play or fold because
of that money you already have in there? $50/$200? Not at
all. That's not your money anymore! It's in a pool of money
to be given to the winner. You have no "stake" to
any of the funds you have already put in the pot. This is
exactly where amateur Texas Holdem players make their biggest
tactical mistake. They think about all the money they have
put in that pot and chase after the pot even though their
mathematical chances of winning are slim to none. The only
stake you might have is totally mental and has no bearing
on hard statistics.
The next step is to use bet
odds and implied odds. That's tougher, because it involves
predicting reactions of other players. With bet odds, you
try to factor in how many people are going to call a raise.
With implied odds, you're thinking about reactions for the
rest of the game. One last example on implied odds...
Say it's another $5/$10 holdem
game and you have a four flush on the flop. Your neighbor
bets, and everyone else folds. The pot is $50 at this point.
First you figure your chance of hitting your flush on the
turn, and it comes out to about 19.1% (about 1 in 5). You
have to call this $5 bet vs a $50 pot, so that's a 10x payout.
1/5 is higher than 1/10, so bet odds are okay, but you must
consider that this guy's going to bet into you on the turn
and river also. That's the $5 plus two more $10 bets. So now
your facing $25 more till the end of the hand. So you have
to consider your chances of hitting that flush on the turn
or river, which makes it about 35% (better than 1 in 3 now),
but you have to invest $25 for a finishing pot of $100. $100/$25
is 1 in 4. That's pretty close. But there's more!... if you
don't make it on the turn, it'll change your outs and odds!
You'll have a 19.6% chance of hitting the flush (little worse
than 1 in 5), but a $20 investment for a finishing pot of
$100! $100/$20 is 1 in 5. So the chances would take a nasty
turn if you didn't hit it! What's makes it more complicated
is that if you did hit it on the turn, you could raise him
back, and get an extra $20 or maybe even $40 in the pot.
As you can see I could imagine
additional scenarios that would make it impossible for you
to call. What you have to do is to master simple outs and
pot odds, and remember that bet and implied odds are just
extended versions of those odds. If you sit and think about
these things while you play, it'll come to you eventually
without any help at all.
Example number one:
playing with a pocket pair.
You start with a pair of Aces
in the pocket. You are holding the top pair. The flop however,
doesn't contain another Ace.
Lesson
1: What are my chances of getting an Ace on the turn?
You need to just figure out
the number of outs and divide it by the number of cards in
the deck. There's 2 more Aces. There's 47 more cards since
you've seen five already (two in your hand and three on the
flop). The answer is 2/47, or 0.0426, let’s say close
to 4.3%.
Lesson
2: No luck on the turn,
how 'bout the river?
Still 2 Aces left, but one less
card in the deck bringing the grand total to 46. What's 2/46?
That's .0434, which is also a little more than 4.3% Your chances
didn't change much.
Lesson
3: Now what if I wanted
to get 4 Aces! What are my chances of that happening?
Since we're trying to figure
out the chances of getting one on the turn AND another one
on the river, and not getting one on EITHER the turn or river,
we don't have to reverse our thinking. Just multiply the probability
of each event happening. Chances of getting that first Ace
on the turn was 0.0426 and the chance of getting a second
Ace on the river would be 1/46, because there would only be
one Ace left in the deck. That's about 0.0217, or 2.2%. To
get the answer, multiply these to together .0426 X .0217 is
about .0009! That's around one-tenth of a percent or one in
one thousand hands that you get pocket Aces to start your
hand - not often.
Lesson
4: Hey, what were my chances of getting a pair of Aces
to start off with anyway?
Lesson
5: What were my chances of getting an Ace on the flop?
Now you do have to "think
in reverse" as in the previous example. Figure out the
chances of NOT getting a Ace on each successive card flip.
First card you have a 48/50 chance (48 non-Ace cards left,
50 cards left in the deck), second card is 47/49, third card
is 46/48. Those come out to .96, .959, and .958. Multiply
them and get .882, or an 88.2% chance of NOT getting any Aces
on the flop. Invert it to figure out what your chances really
are and you get .118 or 11.8%. This will be your chance to
get one more Ace on the flop.
Example number two:
"The straight draw"
You start with a Jack of Spades
and a Ten of Spades. You get a rainbow flop with a Queen of
Spades, a Three of Diamonds, and a Nine of Clubs. You've got
a straight draw.
Lesson
1: What are my chances of hitting it on the next card?
Same as before, but with different
outs. A King or an Eight will complete your hand. There are
presumably four of each left in the deck. You've got 8 outs.
The chance of getting one of them on the turn is 8 over 47,
because there's 47 cards left in the deck. That comes out
to about .170, or around 17%.
Lesson
2: I didn't get it on the turn! What are my chances
now!?
There's still 8 cards left in
the deck that'll help you, but 46 cards left in the deck.
That's 8 over 46. It changes to .174. It's improved to a whopping
17.4%!
Lesson
3: I should of thought about my total chances first,
I'm such an idiot. What are my chances of getting that card
on the turn OR the river?
Once again we'll have to calculate
the chances of a King or Eight NOT appearing, so we can do
it like the last problem (in this case, {39/47} X {38/46}).
Or, since we've already figured out our chances in the previous
two lessons, we can just invert the probabilities and multiply
'em. You had a .170 chance on the turn, and a .174 on the
river. By inverting, I mean subtracting them from one. Now
we've got .830 and .826! Multiply and get .686! That's our
chance of NOT hitting our card at all. So invert it again
and get .314, or 31.4%. So drawing to an open ended straight
draw is 31.4% - now that I like.
Example number three:
"Top two pair"
You get dealt a King of Diamonds
and a Nine of Hearts. The flop is lookin' pretty good for
you with a King of Spades, a Nine of Clubs, and a Four of
Clubs. Top two pair!
Lesson
1: What are my chances of getting a full house on the
turn?
To get a full house, you need
another King or Nine to pop up. There are presumably two of
each left in the deck. So you've got 4 outs. After the flop
there's always 47 cards unaccounted for. 4/47 is around .085
or an 8.5% chance of you getting that boat.
Lesson
2: What are my chances of getting a full house on the
river?
If it didn't happen on the turn,
your chances usually don't change all too much, but let's
check. You've still got 4 outs and now 46 unseen cards left.
4/46 is about .087 or around an 8.7% chance of hitting it
on the river. A .2% difference. Sorry.
Lesson
3: How about the chances of getting the boat on the
turn OR the river?
Like the previous examples,
to figure your chance of something happening on multiple events,
you need to calculate the chance of it NOT happening first.
On the turn it won't happen 43/47 times. On the river it won't
happen 42/46 times. 43/47 is .915, and 42/46 is .913. Multiply
them and get .835, or 83.5% chance of it not happening. Invert
that and you get a 16.5% of getting at least a full house
by the showdown.
Lesson
4: What do you mean by "at least"?
Since we figured the chances
to NOT get dealt a full house, the chances are built in if
the turn and river are two Kings, two Nines, or a King and
a Nine. If you are dealt two cards both of either King or
Nine, it'll be four-of-a-kind and not a King and Nine 33%
of the time. Think of it as being dealt one card then the
other. What are the chances of the first card matching the
second? Whether it's a King or Nine, there will be only one
unaccounted for, but two of the other. That's 1/3, or 33%.
Lesson
5: Then what are my chances of getting four-of-a-kind?
This is a little more abstract.
I hope I warmed you up for this with the previous lesson.
It doesn't matter which card we're banking on. We need to
first get a full house on the turn. According to lesson #1,
the chance of that happening is .085. The chance of getting
the same card we got on the turn is 1/46. There's only one
out, and the usual 46 unseen cards. 1/46 is around .022, or
2.2%. Multiply the two probabilities (.022 X .085) and get
.002 or one-fifth of a percent. It will be Kings half of the
time and Nines the other half.
That
is a lot of information to digest in one shot, but if you
are serious about playing poker to win then you have to become
a master of odds, and you need to review all the odds over
and over again until you know the odds perfectly and you know
the odds in any given situation as well. Like anything else,
practice makes perfect.
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