# Probability and Betting

Understanding the principle of value betting relies upon an appreciation of the probabilistic nature of odds setting.

Probabilities are a way of expressing how likely an event is to occur and can range between zero, denoting that the event will never happen, to 1, which accepts that the occurrence is a certainty.

Real life events on which wagers are made, such as sporting outcomes will have probabilities that lie between 0 and 1, with probabilities closer to zero being less likely to prevail than those that approach 1.

A **coin toss** is an excellent way to convey many of the concepts used in calculating probabilities, expressing those probabilities in a variety of odds formats used by the bookmakers and understanding how the bookmaker’s margin affects the quoted odds.

A fair coin will be equally likely to fall as a head as it is a tail.

So the probability of either a head or a tail is 0.5 and as the number of coin flips increases, the percentage of heads or tails tends towards 50%.

A probability can be easily converted to a decimal odds format by taking the reciprocal of the probability.

Hence a probability of 0.5 is expressed as 1/0.5 or 2.0 in decimal odds format.

Such a wager is unattractive to both the layer and the bettor.

The layer knows that the probability of either a head or tail appearing is 0.5, as does the bettor.

So there is not room for doubt and if the quoted odds are even money or 2.0, the long term expectation is that neither will make a profit.

Consequently, the layer will quote odds for heads and tails that are both shorter than even money and typically odds of 1.9 each of two may be quoted.

Now instead of a one unit winning bet returning the one unit stake and one unit winnings on 50% of wagers, only 0.9 of a unit is returned as winnings, along with the one unit stake.

These less generous prices ensure that in the long term the layers can cover their overheads and secure a profit, while the bettor, despite selecting regular winning bets, will lose money, also in the long term.

In betting parlance, the bet is considered a **poor value** one for the bettors.

Sporting events are decided by a multitude of factors and the true probability of a home win, away win or draw in a football match is not known with the same certainty which exists for a coin toss.

Historical form and team news are just two factors that are typically used to compile football odds for the three possible match outcomes and as with our hypothetical coin toss, bookmakers will then include a margin to reduce their potential pay out on the winning outcome.

However, competition between bookmakers ensures that as well as shortening the prices of the possible outcomes in a football match, they must also remain attractive enough compared to prices quoted by rival firms, so that they can attract wagers from bettors.

Therefore, in situations where there is uncertainty about the true probability of a sporting outcome, but a need to be competitive for the layer, they may occasionally offer a price that still underestimates the probability of a particular result occurring, even after they have applied their margin.

The increase in availability of **innovative statistics and advances in football analytics** has resulted in alternative methods of evaluating footballing match-ups, particularly when identifying sides that may have benefitted or suffered from random, and largely unsustainable bouts of good or bad fortune.

The increasingly low margin markets, particularly on the betting exchanges, such as Betfair, therefore may give rise to value prices compared to match probabilities formed from statistical models, such as those used to power the Infogol app.

To quote a recent example.

High flying Norwich do not appear to have underlying statistical indicators that underpin their strong promotion push.

Infogol’s expected goals figures placed them only eighth best in the Championship after 15 games, rather than the fourth position they actually occupied and they were estimated to have a likely winning probability of 0.47 at home to Leeds in game week 16.

The Betfair exchange price for a home win was 1.86 which equates to an implied probability of 0.54.

Their opponents, Leeds were modelled to have a winning probability of 0.26, compared to an exchange price of 4.80 and an implied winning probability of just 0.21.

If the modelled price for Leeds was a reliable indicator of the Yorkshire side’s likely chances of success in such a match up, the larger returns due to the exchanges under rating of their chances would ensure a long term profit for the model based approach.

Although a Leeds win was still the least likely of the three outcomes in their game with Norwich, the modelled approach was more optimistic about their chances than the market, even with the bookmaker’s margin.

So, while wining wagers would have been less frequent than if siding with Norwich, **the long term pay out** from this value wager on a Leeds victory and others like it would ensure a profit if the modelled outcome was consistently superior to the views of the market.