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.
