Odds and Probabilities
Odds and probabilities are important to the game of poker, they
require a little bit of math, but they are really not that
difficult once you practice with them. The key is to spend some
time working with them so you get comfortable in using them, and
understand what is going on.
Why are Odds and Probabilities Important
Why must we understand odds and probabilities? As I explained in
another article, the skill part of poker, to a large extent,
consists of making good bets and avoiding bad bets. The player
who makes the best decisions will win in the long term. Make the
wrong decisions, and you will lose over the long term.
In a coin flipping example, there is an equal chance of getting
a head or a tail. If you were to put up a dollar for every head
that turned up, and your friend put up a dollar for every tail
that turned up, you would have an even bet. Since the chances of
a head or tail on any coin flip are equal, your odds are also 1
to 1. One time you will win, and one time you will lose. In this
example, both you and your friend expect to win the same amount
over time, $0. This is neither a good bet nor a bad bet, but
neutral. You shouldn't be interested in taking the bet however
because the best you can do over time is break even.
On the other hand, if you put up $1 for every head, and your
friend put up $2 for every tail, you would have an edge. The
chances of flipping heads and tails will still be even, but you
get paid more when you win, so over time you will make a profit.
This is a bet you should take since you expect to make a profit
over time. If you had to put up $2, and your friend put up only
$1, you would expect to lose money. This would be a bad bet for
you.
In a coin flipping example, the choices are easy to understand,
since there are only 2 possibilities. This is not always the
case. In poker the choices are much more complicated, which is
why it is important to understand odds and probabilities, in
order to make good decisions.
Probability
Probability is the likelihood that something will happen. For
instance, when you hear the weather report in the morning, and
the weatherperson tells you that there is a 20% chance of rain
they are saying that the probability of rain is 20%.
Some important concepts to understand here are that if there is
a 20% probability that it will rain, there is an 80% probability
that it will not rain. Probabilities can not add up to more than
100%, and the sum of all of the various possibilities must add
up to 100%.
In simple cases like a coin flip, or the chance of rain, where
there are only 2 possibilities, the 2 probabilities will add to
100%. In some situations however there will be more than 2
possibilities. If we only calculate some of the probabilities,
they will not add up to 100%, because we did not consider all of
the possibilities, but those possibilities still exist, and must
add up to 100% in total.
Another way to write the same information is to say that there
is a .2 probability of rain, and that there is therefore a .8
probability that it will not rain. The total probability can not
be more than 1, and once again all of the possibilities must add
up to 1.
Odds
Odds are a different way of expressing the same information, but
in a way that is often more applicable to poker and other
gambling games.
While probability is expressed as a decimal number, or a
percentage, odds are expressed as 2 numbers separated by a colon
such as 5:1. By convention this notation indicates that the odds
are 5 to 1 against the event occurring.
There are different ways of saying the same thing, and of
explaining what the numbers mean. In the example, let's assume
that the event we are interested in is getting 1 particular card
that we need in order to make our hand. The notation tells us
that 5 times we will fail to get the card we need, and 1 time,
we will get the card we need. Using that same example, we will
get the card we need 1 time in 6 attempts, or 1/6.
Working with Odds and Probability
Note that although probability is normally stated as a
percentage, or a decimal number, percentages and decimal numbers
are simply fractions expressed, or written, in a different way.
For instance, 1/6 is the probability of getting the card we
need. If you divide the 1 on top, by the 6 on the bottom, you
get .167, or 16.7%. All 3 of these numbers mean exactly the same
thing, there is a probability of 1/6, or .167, or 16.7% of
getting the card we need.
Putting it all together, 5:1 means losing 5 times for every 1
win, winning 1 time out of 6 attempts, the probability of
getting the 1 card is 1/6, .167 or 16.7%. The probability of not
getting the card you want is 1 - .167, or .833, or 83.3%. Once
you know the probability of getting the card, and the
probability of not getting the card, you can put that
information into the form of odds. In our example that becomes
83.3:16.7 against getting your card.
You normally reduce odds to the form X:1 to make comparisons
easier. To do that, you simply divide both numbers by the number
on the right. i.e. in the example 83.3:16.7 you divide 16.7 by
16.7 to get 1, and then divide 83.3 by 16.7 to get 5, giving you
5:1, which is exactly where we started.
Of course if you do the math you will see that I rounded the
number off in all cases since numbers like .16666666666 are
difficult to work with, and for our purposes, .167, .833 and 5
are plenty accurate enough.
Going back to the weather example from the beginning, there is a
20% chance of rain, which means that there is an 80% chance that
it will not rain. Putting these numbers in the form of odds, it
is 80:20 against it raining. Simplifying, divide both sides by
20 and you get 4:1 against it raining. You can put this back
into the form of a probability by adding the 2 numbers together
and then putting the right number on top, i.e. 4 plus 1 is 5,
put the 1 from the right side on top of that and you get 1/5.
There is one chance in 5 that it will rain. To express the
fraction as a decimal number, divide the number on top by the
number on the bottom, i.e. 1 divided by 5 and you get .2. To
express that as a percentage, multiply by 100 and you get 20%
chance of rain. Right back with the number we started with,
because they are all ways of saying the same thing.
Why use Both
Stating the situation in the form of odds, as in 5:1 gives us a
clearer picture of where we stand than saying we have a 16.7%
chance of getting the card. As well, it gives a more complete
picture since, for the probability we want to know both that
there is a 16.7% chance of getting the card and an 83.3% chance
of not getting the card.
Odds can not be used in all situations however. For instance, on
the first card you are dealt, the odds of getting an Ace are
12:1, the odds of getting an Ace on the second card, given that
you got an Ace on the first, are 16:1. If you want to know the
odds of getting a pair of Aces however, you can not calculate
them directly from the odds, you must use probabilities.
Using probabilities to do this, there are 4 Aces out of 52
cards, so the probability of getting an Ace on the first card is
4/52 or 1/13. The chances of getting an Ace on the second card
are 3 Aces, since we already have 1, in 51 remaining cards,
which is 3/51 or 1/17. You can then multiply the 2 probabilities
to get the answer.
You can do this in 1 of 2 ways. You can multiply the fractions
4/52 * 3/51, or 1/13 * 1/17, to get 12/2652 or 1/221 and then
convert to odds. i.e 2652-12:12, is 2640:12 is 220:1, or 221-1:1
is 220:1.
You can also convert each of the fractions to decimals, 4/52 ~
.077, and 3/51 ~ .059, then multiply .077 * .059 ~ .0045,
convert to a percentage by multiplying this number by 100 and
there is a .4525% of getting a pair of Aces as your first 2
cards. Since there is a .4525% chance of getting a pair of Aces,
there is a 100 - .4525 = 99.5475% chance of not getting a pair
of Aces. The odds against getting a pair of Aces on the first 2
cards are 99.5475:.4525, simplifying, we divide both sides by
.4525 and we end up with 220:1, the same answer.
Note that when doing a series of operations such as above, you
can't round the numbers off until you complete all of the
calculations or it will significantly affect your results. I
used numbers such as .077 above instead of typing out the entire
long decimal number, but I used the actual numbers in the
calculations.
Of course, trying to do this math at the table would not be
practical, so for many common situations we memorize the odds,
or probabilities. For instance, the odds of someone having any 1
specific pair as their first 2 cards are 220:1. i.e. it is 220:1
that they will have KK, 220:1 that they will have QQ etc. In
order to make memorizing easier, I will provide tables of many
common odds and probabilities in later articles.
As we will see in the next Odds related article, there are a
couple of more good reasons to use odds instead of probability.
One is that odds are much easier to calculate while sitting at
the table. The other is that the odds can be used directly in
deciding if we have a good bet or a bad bet.
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