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 BackJack 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.
Therefor
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
