You’d be forgiven if you’ve never heard of Thomas Bayes, an 18th century reverend, philosopher and mathematician, but his Theorem has become an integral part of betting theory and strategy. At its heart it is a method of calculating probabilities for outcomes based on prior knowledge and statistics, which is as good a description of smart betting as you will find.
The Englishman’s eponymous theorem can be applied to all manner of scenarios to make conditional-based probability decisions. This could be anything from assessing the chances of rain tomorrow to the likelihood of a borrower defaulting on a loan. More importantly – for us at least – Bayes’ Theorem can be applied to sport, including horse racing.
As an equation, Bayes’ Theorem looks like this:
P(A|B), which is the answer, is the conditional probability of A occurring given B is true.
P(B|A) is the probability that the condition B will be present given A is true.
P(A) and P(B) are the probabilities of A and B without regard to one another.
Got it? No? Don’t panic. This all probably looks and sounds confusing but stick with me because it’s actually fairly straightforward.
A racing rivalry
Let’s imagine a scenario where two horses are set to race one another:
Horse A has won seven of its previous 12 clashes with Horse B
Horse B has won five of its previous 12 head-to-heads with Horse A
This means we can estimate Horse A’s chances of winning as 7 ÷ 12 (0.583) or 58.3%. Horse B’s estimated probability of winning is 5 ÷ 12 (0.416) or 41.6%. So who would you bet on?
You’d probably be more inclined to back Horse A because you think it is more likely to win given the head-to-head (H2H) record. However, let’s muddy the waters slightly by saying that four of the 12 head-to-heads were on soft ground and three of Horse B’s five victories were on soft ground while one loss was also on soft.
Ahead of this latest encounter the going is described as ‘soft’ following sustained rainfall. So we need to combine these pieces of information to come up with probabilities. This is the essence of Bayes’ Theorem. We take an initial probability (the H2H record) and add in a condition that could alter it and rework the probability.
Remember, Horse B won three times when the going was soft and won twice on good ground for five wins in total and there have been four soft ground races of the 12. Using Bayes’ Theorem to calculate the probability of Horse B winning when the going is soft, we know:
P (soft ground | winning) = 3 ÷ 5 = 0.6
P (winning) = 5 ÷ 12 = 0.417
P(soft ground) = 4 ÷ 12 = 0.33
Expressed using the Bayes’ Theorem equation, it is as follows:
Which is: P (Horse B winning ÷ soft ground)
So the probability of Horse B winning on today’s soft ground is 0.6 x 0.417 ÷ 0.33 = 0.75. This raises its chances of winning from 41.6% to 75%. I think you’ll agree it’s a significant jump and now means Horse B is the clear favourite (1.33 in true decimal odds) despite having fewer previous victories.
What you are doing is adding a condition to an existing probability and seeing the impact this has on the predicted outcome. It’s something you do all the time in real life without knowing it. Say you hear a vehicle outside, what are the chances it’s a car and what are the chances it’s a milk float? Now how about we say it’s 4AM? How has that condition affected your probability?
Obviously horse racing is way more complicated than this and packed with a head-spinning array of variables, but Bayes’ Theorem allows us to make more informed decisions with just a little bit of work.
What we’re looking to do is apply some mathematical rigour to ‘gut feeling’. We all know that a horse suited to conditions or on a favoured ground has an increased probability, but by how much does that probability increase? This is where Bayes’ Theorem comes in.
You can either use the theory in its purest sense above for individual horses or you can look more broadly for something called “likelihood ratios” that use the same theory but a slightly different calculation. The basic starting point is all horses have an equal chance and we then adjust based on these factors, so perhaps it’s most applicable for handicap races.
So let’s imagine we are focusing on an eight-runner flat race at Sandown Park and we are going to say every runner has an equal chance of winning. This means they all start with an implied probability of 0.125 (12.5%) or decimal odds of 8.0.
So, let’s set our new conditions. This time, we could choose to focus on horses that won last time out over today’s trip of seven furlongs. Delving into historical data we have gathered on 9,391 runners throughout the flat season so far, it shows:
Applying Bayes’ Theorem we’re trying to work out the likelihood a horse that won last time out over 7f will win today. The calculation is slightly different to the earlier one and uses this model:
Horses where factor applies:
This is Horses with factor that won X (total last time out losers) ÷ Horses with factor that lost X (total last time out winners)
Horses where factor does not apply:
This is Horses without factor that won X (total last time out losers) ÷ Horses without factor that lost X (total last time out winners)
Those horses where this factor is applicable would rise to 0.153 (0.125 x 1.23), or 15.3% probability. Expressed as decimal odds, these runners would shorten to 6.53, which is a pretty hefty fall from odds of 8.0. Meanwhile, those runners where the factor is applicable would dip slightly to 0.1225 (0.125 x 0.98) or 12.25%.
Expressed as decimal odds, it is a slight rise from 8.0 to 8.16. You can apply all sorts of factors and once you’ve translated the probabilities into odds to identify horses that are value. In other words, horses available to back at longer odds than your calculations suggest.
It’s also recommended that, if possible, you apply factors that have the greatest influence over whether probabilities increase and decrease. For instance, applying a factor with a result of 1.01 or 1.02 is going to have an inconsequential impact so probably should be ignored.
Bayes Theorem isn’t a magic trick to finding good bets, but it can help you make more informed and accurate decisions around conditional probability. In other words it is a more focused approach to assessing the impact of the conditions of an event. And, of course, it’s not just horse racing; Bayes’ Theorem can be applied to pretty much any event.
For instance, you could use it to determine the probability of a certain tennis player beating an opponent who dominates the H2H record but the former has won most matches on today’s surface of clay. It all means we can harness the genius of a man from almost 300 years ago to try to predict the future – and hopefully make some money in the process.