# What is a Bet? How to Calculate Mathematical Advantage?

### What are â€œoddsâ€ and what is a â€œbookmakerâ€?

Bookmakers are financial institutions which offer bets for sale, professionally. Returning to our example, instead of betting with a friend one would make such an agreement with a bookmaker on the outcome of the coin toss.

It is obvious enough that any company wishing to make a living from selling bets will always guard against accepting those which will lose on a long-term basis, in other words, bets which do not contain a mathematical advantage (for example, an advantage like paying out 90 Cents in exchange for accepting a one Euro stake on a 50/50 chance every time).

In the case of our coin toss the mathematical advantage is computed as follows:

There were two outcomes with exactly the same probability and if the coin is not damaged or weighted/biased in any way, there should theoretically be 50 times one result and 50 times the other. A coin does not have a memory and so, the likelihood for each new throw is always exactly 50% for â€˜headsâ€™ and 50% for â€˜tailsâ€™ time and time again, ad infinitum.

â€˜Oddsâ€™ is the price the bet on the desired outcome is offered at for sale by the bookmaker, showing what will be won by the player for a unit stake of 1.

If one wants to express the ‘chance’, or ‘probability’ or ‘likelihood’ of â€˜headsâ€™ or â€˜tailsâ€™ in terms of odds, then the â€˜fairâ€™ or â€˜zeroâ€™ odds (i.e. the break-even odds) for this example would be 2.00 (please note: there are several different ways of displaying odds, but all of them express the same meaning (like different languages); it’s just that different parts of the world prefer different notations. Soccerwidow uses so-called ‘decimal odds‘ which are favoured in continental Europe, Australia and Canada; however the UK is more familiar with fractional odds, and America with Moneyline odds – read more in Wikipedia.).

Decimal odds are arrived at through dividing 1 (unit stake) by the probability that an event will happen. In our example of the coin toss, the probability is 50% as there are only two possible outcomes, ‘heads’ or ‘tails’.

â€˜Fairâ€™ or â€˜zeroâ€™ means that these odds exactly mirror the probability of the outcome and, in the long run, reflect that one neither wins a profit nor suffers a loss when betting often enough on the same outcome.

In our example however, the opponent (i.e. the bookmaker) paid out only 90 Cents on a bet of one Euro stake which had a 50/50 chance to win or lose. Therefore, the bookmaker offered the bet on â€˜headsâ€™ to win at the following odds (price):

1 â‚¬ stake plus 0.90 â‚¬ pay out = 1.90 â‚¬ total pay out = 1.90 odds

Odds of 1.90 convert into a probability of:

1 divided by 1.90 = 0.5263 or 52.63%

This means that odds of 1.90 represent a calculated likelihood of an event occurring at 52.63%. In other words, the bookmaker sold the bet at a price which corresponds to a probability of 52.63%. We have already seen that the coin toss has an actual probability of only 50% and so, the bookmaker has priced the bet with a â€˜mathematical advantageâ€™ on his side (in casinos this is called the â€˜house advantageâ€™). To compute this:

52.63% (bookmakerâ€™s probability) divided by 50% (actual probability) equates to 105.26 (%), or 5.26% above the â€˜levelâ€™ or â€˜fairâ€™ price.

Alternatively, one can calculate the mathematical advantage by:

2.00 (calculated fair odds) divided by 1.90 (offered odds) = 1.0526 (5.26%)

A third variation of computing the mathematical advantage:

In our example, the player lost 50 x 1 â‚¬ = 50 â‚¬ (in other words, paid out 50 â‚¬ to the bookmaker); the bookmaker lost 50 x 90 Cents to the player = 45 â‚¬ (winnings of the player).
50 â‚¬ plus 45 â‚¬ = 95 â‚¬.

â€˜Fairâ€™ would have been if both parties had paid out 1 â‚¬ equally since the probability was exactly 50/50; 50 â‚¬ (player to the bookmaker) plus 50 â‚¬ (bookmaker to the player) = 100 â‚¬.

100 â‚¬ (at ‘fair’ odds) divided by 95 â‚¬ (at actual odds played) = 1.0526 (5.26%)

Whichever way it is calculated, the mathematical advantage to the bookmaker in our example was 5.26%. On bets with the mathematical advantage in his favour, the bookmaker will always win money from the player in the long run.

Last Update: 23 January 2012

### 3 Responses to “What is a Bet? How to Calculate Mathematical Advantage?”

1. 1 September 2012 at 2:20 pm #

Hi Soccerwidow.
I read most of your articles about odds calculation and its all great stuff and just continue with good work. I will try to recommend you to as much people as I can 🙂
let s say I found two matches whit higher odds than zero odds, first 2.4(bookie odds) / 2.25(true odds) = 6.6% advantage, second 1.65(bookie odds) / 1.5(true odds) = 10% advantage. So in this case if I play multiples I get a mathematical advantage of 17% (2,4*1,65 / 2,25*1,5) right? And if would continue to play like this I would need to adjust my stake so the winning amount is the same in every case?
Thanks.
Best regards

• 1 September 2012 at 5:51 pm #

Hi Betman,

Regarding staking, I would always recommend the fixed win/ fixed win plan. Meaning adjusting the stake that you don’t exceed a fixed amount of risk (lay bets), or limit your winnings to a fixed amount.

However, I would advise against multiples. It is already difficult enough to predict individual matches.

2. 5 February 2012 at 12:00 pm #

hi there,
this is really useful stuff youre posting here. Im glad i have came across your site. Im looking forward to see some more advanced articles 😉

br/bester