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
