In a recent article (A
Low Flopped Flush), I did some basic probability calculations to
illustrate the likelihood of a player holding 4 2 on a flop of three
clubs being beaten by a higher flopped flush, when eight other players saw
the flop. This article will further that discussion by introducing some
basic probabilities applicable to Texas Holdem and delve into some of the
simplest probability methods.
Starting Hands
Every Texas Holdem player, regardless of experience, has encountered this
most basic probability aspect of the game. Assuming a standard pack of 52
cards, how many possible hands can be dealt?
There are two ways to approach this problem. The more accurate method
(using the formula for combinations) is, in this case, more
time-consuming, because the calculation can be made with simple arithmetic
very quickly. Starting with 52 cards, perfectly shuffled, you can receive
any of the 52 on the first card. The second card can then (obviously) be
any of the remaining 51.
Multiplying 52 (possibilities for first card) x 51 (possibilities for
second card) yields 2652 different ways to deal two cards out of a
fifty-two card deck. Since the order that the cards are received doesn’t
matter, we divide this number by two to come up with the actual number of
possible starting hands, which is 1326. We do this so we don’t count each
possible hand twice, since A 7 is the same as 7 A.
Known Versus Unknown Cards
You may ask, “but what about the other nine players at the table?” This is
a fair question. If you receive the first card dealt, then nine other
players receive their first card, isn’t it true that there are now only 42
possibilities for your second card? This would be true if the nine
opponents received their first hold card dealt face-up, so you would know
a total of ten cards after the first round of dealing. The fact is that
you only know one card, even though ten have been dealt. There are
fifty-one unknown cards as far as you’re concerned, and the unknown cards
are all you have to work with.
To illustrate the significance of unknown cards, here is an example. Say
you receive A K as your hole cards. You raise first in to the pot. Seven
other players behind you fold and the big blind calls. The big blind is
now out of chips, so he is all-in. Both of you expose your hole cards, and
he turns up Q Q. The flop comes 7 4 8. What are your odds of winning?
Since you have no straight or flush possibilities, your only hope is to
pair one of your hole cards on the turn and river (avoiding one of the
queens, which would leave you drawing dead). You have six outs, and with
two cards to come, you have about a 25% chance of winning. You are a 3-1
underdog. After the hole cards are exposed and the flop comes, you know 7
cards. There are 45 unknown cards. On the turn, you have a 6/45 chance of
catching one of your outs. If you do not catch on the turn, you then have
a 6/44 chance on the river. We use 44 unknown cards in the river
calculation because one more card (the turn) is now known. Adding
(6/45)+(6/44) gives us 0.27, which approximates the actual 25% chance of
winning. (The 0.27 is reduced somewhat in actuality by the chance that a Q
comes.)
Now, let’s look at the same hand with slightly different circumstances.
After you raise first in with A K. The next seven players fold their
cards face up, and six of them had either an ace or king. Now all six of
your outs are known to be out of play. The big blind calls, with many more
chips left in his stack. After the same 7 4 8 flop, the big blind bets.
Can you call? Now you know for a fact that your six outs to the AK are no
good. The flop gave you no draws to a flush or straight, so your only hope
is that the opponent cannot beat your AK high hand. If your opponent has
at least a pair after the flop, your chance of winning is zero. The
difference between known cards and unknown cards is clearly significant.
What About Aces?
There’s little doubt that we’d all select pocket aces as our starting hand
of choice. So what are the odds of being dealt “American Airlines?”
There are four different aces, and six different ways of receiving the
coveted pocket pair. If you receive the A first, there are three
possibilities for your second ace. If you receive the A first, you have
two options for your second (we don’t include the
here since the A A
combination was already counted). Finally, with the A you have one more
possibility.
Proper Terminology
This is a good place to demonstrate the proper way to express odds and
probabilities. Of the 1326 possible starting hands, six of them contain
two aces. So your probability of receiving any AA is 6/1326, or 0.00452.
Probability is always expressed as a number between 0 and 1, and
represents the chance that a specified result will happen. A zero
probability means there is no chance of the event happening. A one
probability means the event is certain to happen.
Probability can also be expressed as a percentage, as in “you have a
0.452% chance of being dealt pocket aces.” The percentage will range from
0 to 100. Converting the numerical probability to/from the percentage
probability is simply a matter of multiplying or dividing by 100.
The term “odds” is similar to probability, but odds are a way of
expressing the chance that a specified result will not happen. In the case
of pocket aces, there is a 1320-to-6 chance that you will not be dealt
aces. Dividing 1320 by 6 simplifies the expression, yielding 220-1 odds
against being dealt aces.A related article,
Introducing Statistical Calculations, shows how to use
a more complex statistical computation to determine some probabilities.
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