What is a Bet? How to Calculate Mathematical Advantage?

What are “odds” and what is a “bookmaker”?

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 ‘heads’ and 50% for ‘tails’ time and time again, ad infinitum.

‘Odds’ 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 ‘heads’ or ‘tails’ in terms of odds, then the ‘fair’ or ‘zero’ 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’.

‘Fair’ or ‘zero’ 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 ‘heads’ to win at the following odds (price):

1 € stake plus 0.90 € pay out = 1.90 € 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 ‘mathematical advantage’ on his side (in casinos this is called the ‘house advantage’). To compute this:

52.63% (bookmaker’s probability) divided by 50% (actual probability) equates to 105.26 (%), or 5.26% above the ‘level’ or ‘fair’ 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 € = 50 € (in other words, paid out 50 € to the bookmaker); the bookmaker lost 50 x 90 Cents to the player = 45 € (winnings of the player).
50 € plus 45 € = 95 €.

‘Fair’ would have been if both parties had paid out 1 € equally since the probability was exactly 50/50; 50 € (player to the bookmaker) plus 50 € (bookmaker to the player) = 100 €.

100 € (at ‘fair’ odds) divided by 95 € (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.