Gambling
odds and probabilities
(Please read this in full - it is a worthwhile investment
of your time)
What has statistical probability
got to do with gambling? In a word, everything. If gamblers
had even a modest understanding of probability then
the world's casino's would all be empty. In this article
I will explore the basic statistical probability you
will need to know before you start betting either on
the horses or on other forms of gambling.
Introduction - Odds ain't
Odds
Most gamblers are comfortable
with the concept of odds because we are very interested
in what the pay-out will be for any particular wager.
Many people however, fail to realise that odds are really
a measure of probability and what we should be more
interested in is if the odds offered (the pay-out) correctly
represents the statistical probability of the outcome
we are about to invest in. The words probability and
odds are often used interchangeably since 'odds' is
the language spoken by gamblers but always remember
that when you say odds you mean probability!
To demonstrate what I mean
when I say 'odds ain’t odds' consider a coin toss.
Assuming the coin is without any manufacturing faults
we all know that if it is tossed thousands of times
the number of heads tossed will be about the same as
the number of tails. The probability of heads is equal
to the probability of tails.
In betting terms this is
an even money bet, or a ratio of heads to tails of 1:1
and so the odds are 1/1. These odds are also called
the 'true odds' because the pay-out represented by these
odds correspond to the actual probability of the event
happening. As a percentage the probability of tossing
a head is 50%. Therefore if you win $5 when a head is
tossed and lose $5 when a tail is tossed then at best
you should only hope to break even in the long run.
Along the way you might get ahead for a while or get
behind for a while but over time you expect to break
even.
Is there a way to make
money from this seemingly pointless bet? Suppose you
found someone who was prepared to accept less than $5
(even money) for a correct call of the coin toss? In
other words find a player who will accept a pay-out
of $4 and not the $5 the 'true odds' of the bet would
indicate. Of course if the player gets it wrong, you
keep the full stake of $5!
Instead of breaking even
over thousands of tosses you will steadily send the
other player bankrupt because what you are really doing
is pocketing 20 percent ($1 out of $5) of the other
players money every time he wins. The longer the player
bets the more money he must lose. In betting terms you
are offering odds of 5/4 ON (odds on) when, as you know,
the 'true odds' is even money. In racing terms the hapless
punter is 'taking under the odds'. The odds offered
are called the ‘betting odds’ or ‘gambling
odds’. The true odds represent the statistical
probability of the outcome you are investing in.
The ‘true odds’
are fixed for any particular bet but you can (and will)
be offered any odds at all. The only predictable relationship
between statistical probability and gambling odds in
general is that any sensible gambler will try to offer
you odds that are below the true odds dictated by statistical
probability.
This is very important
so one more time now and say it after me. The ‘true
odds’ are fixed for any particular bet but you
can, and will, be offered any odds at all.
Who would be silly enough
to take a bet that doesn't pay out the 'true odds' you
may ask? Well just wander into a casino and watch those
hapless souls donate their money to the casino owners.
How many of us can say that we have never taken 'under
the odds' on a racehorse? When was the last time you
brought a lotto ticket? The short answer is that we
all have at one time or another. A more appropriate
question to ask is why is it that so many people are
quite happy to go through their whole life betting under
the odds?
In my opinion it is a national
scandal that in casino's people are playing games that
they simply cannot win, the longer they play the more
they MUST lose. It is literally a licence to steal money
from those people unaware of the mathematical futility
of their endeavour.
For you, the savvy punter
make sure you know and understand the difference between
‘true odds’ and 'taking under the odds',
study a few casino games if you still think you can
win at the casino. By the way if you must go to the
casino then only play Back Jack as this is the only
game where the house won’t have a significant
edge.
So how do we make our money?
In the casino the odds are fixed and you either bet
the percentages offered or have a cup of coffee but
in horse racing the odds are fluctuating over the course
of betting for all sorts of reasons, many of which are
totally unrelated to the statistical probability of
the horse’s winning chances.
Continuing with the coin
toss example what if someone offered us a win of $6.25
on heads for a $5 stake? This is called betting 'over
the odds', an over or an overlay. If you can put yourself
in this position then you will win, the longer you play
the more you will win. In betting terms you are getting
odds of 5/4 for an event with 'true odds' on evens,
or 1/1 if you prefer. The other punter is really paying
you a bonus of 25 percent every time you win. Study
the example I have used until you know the difference
between getting 'over the odds', 'under the odds' and
'true odds' because this is the single key to the success
or otherwise of your betting future.
I don’t want to introduce
too many new ideas at this stage but I should point
out that the 25 percent ‘bonus’ in my example
is not to be confused to the percentages that punters
talk about in the context of probability. My 25 percent
was just a calculation based on the stake money I used
($6.25) and the amount of money that I would win ($5)
. The ‘bonus’ is just $1.25/$5 or 25%.
If you were to consider
my example in terms of percentages related to probability
then what is happening is that for an even money bet
you expect to win 50 percent of the time. For a bet
of 5/4 you expect to win 44 percent of the time and
for a bet of 4/5 (or 5/4 ON if you prefer) you expect
to win 56 percent of the time. So the actual fluctuations
in terms of probability between these bets is only 6
percent.
In my example you can see
that if someone is offering me odds of 5/4 that I only
need to win 44 percent of the time (or 44 tosses in
100) to break even, and of course I expect to really
win 50 tosses out of 100. This is the simple reason
that I expect to win over a period of time and once
you understand this concept you will never play another
casino game again, ever.
If you don't feel comfortable
talking in terms of odds and percentages just yet the
important point to grasp is that if the pay-out when
you win is less than the true odds would indicate then
you will never win the game and the longer you play
the more you will lose. Sure you may get ‘lucky’
and get ahead for a while but in the long run you will
lose. The probability of winning in my example is the
SAME for both players but if the pay-out can be manipulated
by either player then one or the other will make money
and the other MUST lose money over a period of time.
How does this apply to
horse racing?
Most people think that
horse racing is about picking winners. Indeed I used
to say to my percentage punting friends "you won’t
go broke backing winners" and didn’t pay
too much attention to the odds simply because I took
the view a winner is a winner at any price. However
the flaw in my logic is that ultimately there are no
good things on the race track and so the odds you take
for your winners is just as important in racing as it
is in the coin tossing example. In the long run if a
bookmaker can get you to take 2/1 about a horse that
should be 5/2 then he will beat you.
Eventually I saw what all
these ‘percentage’ players were on about.
A favourite saying of these punters is ‘good things
come and go but percentages go on forever’ or
another one is ‘you can’t beat a race but
you can beat the races’. I interpret this to mean
that when a horse wins it can be seen as a random event
from race to race but with a probability that can be
measured over many races and hence as a percentage over
a period of time.
It doesn’t really
matter if your next bet gets up (just as in the coin
toss) as long as over a period of time the percentages
are in your favour. If you plan to bet over hundreds
of races then you must use a system that is designed
to win over hundreds of races and certainly not rely
on putting large amounts on this weeks ‘good thing’.
You will, obviously want
to back the horses with the highest probability of winning
but only at better odds than the ‘true odds’.
The art of horse racing is being able to determine what
horses are over the odds and what horses are under the
odds and not simply picking winners. This of course
raises the issue of how do you work out the odds (probability)
of a horse in a race? A coin toss or a roulette wheel
is easy but a horse race?
Well the answer is we can't,
not exactly anyway, but many astute punters can analyse
form to the extent of getting a good approximation of
the probability of each horse in a race. How people
do this and how well they do it is a topic for another
day.
Working out the probability for a single event
Working out probability
can be simple or quite difficult depending on the situation.
In the simple case you need to work out just two things,
how many outcomes are possible and which of these outcomes
are successful for the wager you are making. To calculate
the probability of success you simply divide the total
number of successful outcomes by the total number of
possible outcomes.
So if an outcome has ‘n’
ways of occurring and only one outcome counts as a success
then the probability of the event happening is simply:
p(Success) = 1/n
A probability of one means
that an event is certain to happen while a probability
of zero means the event us certain not to happen. There
are a couple of useful rules like:
p(Success) + p(Failure)
= 1
(or in words it is certain
that the even will either occur or not occur, agree?)
and so once you know either
the probability of success or failure you can work the
other out using the formula:
P(success) = 1 - P(failure)
P(failure) = 1 - P(success)
As an example lets work
out the probability of drawing the ace of spades from
a pack of cards and then convert this number to odds.
The total number of outcomes possible, ‘n’,
is 52, since there are 52 cards in a pack. There is
only one successful outcome so the probability is:
1/52 = .019 or approximately
2 percent.
Thinking in terms of percentages
is often useful. If this percentage was for a horse
in a race you would know that for every 100 races you
would only expect a horse with this probability to win
twice. A long time between drinks don't you think?
Converting Odds to Probability
Now let's solve one of the great mysteries for many
a punter, converting odds to probability. But before
we do a word about odds. Odds are simply the ratio of
the losing outcomes (or chances) to the winning outcomes.
Bookmakers usually express
odds as odds against winning. So a 10/1 horse has 10
chances of losing and only one chance of winning and
as a ratio this is 10:1. A 6/4 bet would have 6 chances
of losing and 4 of winning and of course an even money
bet, 1/1 has one chance of winning and one chance of
losing.
Remember odds are really
a ratio and should be expressed as 10:1, 6:4, 1:1. The
‘:’ (colon) is usually replaced with a ‘/’
(slash) and I can only assume that this is for the convenience
of bookmakers in working out what their pay-outs will
be. For the purposes or converting odds to probability
the '/' does not work as the divide symbol so mentally
replace it with a ':' and you will find life much easier.
Now a special case is when
a horse has more chance of winning than losing, eg 4/6,
4 chances of losing a six of winning of winning. These
horses are called 'odds on’ an usually appear
in red on the bookmakers board. Just to confuse you
further most people just say 6/4 ON. If you see this
just convert it in your head back to 4/6,or more correctly
4:6.
As we have discussed a
horse showing odds of 10/1 has 10 chances to lose and
only one chance to win (remember bookies odds are odds
against an event happening). Now this is where knowing
that the odds are really a ratio is important. 10/1
is really 10:1 and so you have 10 chances of losing
and 1 chance of winning. The total number of chances
is 11.
Therefore the probability
of winning is 1 chance in 11 or 1/11 = .09 or 9%. Many
people get this wrong because when they see 10/1 the
think that they have one chance in 10 of winning but
really it is one chance in 11. Once you treat odds as
a ratio you never make this mistake again.
So if odds are expressed
as ‘odds/1’ then as a ratio this is ‘odds:1’
and the total number of possible outcomes, n is then
‘odds+1’.
probability = 1/n and as
a percentage = (1/n)*100
Another example, odds of
4/1 (or as a ratio 4:1)
n = 4 + 1 = 5
Probability = 1/5 = .2
or 20%
Most people just add one
to the quoted odds and divide this number into one.
It is a simple formula and by all means use it but always
remember odds are a ratio. In the real world examples
understanding this will be a great help.
Converting Probability to Odds.
Again before we simply
use a formula and forget about the subtleties lets work
out the odds at least initially using a method that
gives you some insight into what you are doing. Given
the probability of drawing the ace of spades is 1/52
how do we work out the odds you would bet about doing
this?
First ask yourself how
many chances, or ways if you prefer, are there to win?
In a horse race this will always be one and in our card
example this is also 1. Then ask how many ways are there
to lose? In the card example this is 51 (since 1 card
is the winning card, 51 cards are losing cards). Now
you recall that I have stated that odds against is simply
the ratio of losing to winning outcomes and so:
Odds = 51:1 as a ratio,
(51 chances to lose and only 1 to win).
or 51/1 as you would see
on the bookies board.
If you prefer to use a
simplified formula here it is:
odds = (1/prob) -1
and call the result ‘something’
:1 or ‘something/1’ whichever you prefer.
For instance suppose you
have a probability of ¼ or .25.
Using .25 the odds are:
odds = (1/.25) -1 = 4-1
= 3
and so the odds are 3:1
or 3/1.
When you want to avoid
rounding errors (eg. 1/52 is really 0.0192307... and
not just .019) then use the 1/n representation for probability
in the above calculation and not the rounded decimal
probability.
Ie. P = ¼ instead
of .25, so
odds against = (1/(1/4))
-1 = 3 and odds are 3/1 as before. For most practical
uses in horse racing the rounded decimal representation
of probability is close enough.
Conclusion:
Converting between odds and probability is easy once
you know a few simple rules. Since as I have already
stated, the only predictable relationship between statistical
probability and gambling odds in general is that any
sensible gambler will try to offer you odds that are
below the true odds dictated by statistical probability
you owe it to yourself to be able to do these calculations
for yourself before embarking on any serious attempt
to make money from your chosen form of gambling.
This article is copyright
Doug Robb 1996. All rights reserved. May be copied freely
for personal use and yes you can put it up on your web
page providing this copyright notice stays in tact
