It’s possible that one man has had more impact on the gambling industry than any other. His name is Thomas Bayes. Many people have written, without evidence, that he was a gambler, but he is best known for a rule he formulated in the 18th century.
Bayes, a Presbyterian minister, was born in 1701 in London and during his life, he published only one mathematical work, a defence of Newton’s calculus. However, in 1742, he was elected a fellow of the Royal Society, which suggests he was recognised as a serious scholar by his fellow scientists. But the work that he is best known for was published in 1763, two years after his death, by his friend, Richard Price. Bayes’s paper was edited quite heavily by Price before he circulated it.
The mathematical theory contained in Bayes paper, “An Essay Towards Solving a Problem in the Doctrine of Chances”, was fairly controversial and even disparaged as subjective and impractical; apart from initial interest, it was largely ignored until the 1970s when the usefulness of Bayes theorem in statistical analysis became apparent, thanks to the widespread availability of computing power. What was Bayes theorem? Simply put, it allowed you to update your belief in the likelihood of something based on new evidence becoming available; what is the probability of event A given the new evidence B?
Although a very simple concept and something we humans do all of the time, we tend to reframe probabilities with poor rational thought. Imagine you had a bag of tiles (like Scrabble tiles) all of the same size and you know that 70% of the tiles in the bag are red and 30% are black. Pulling a tile from the bag without seeing it, you can say that there is a 70% chance that the tile is red.
However, you are then given the information that 80% of the red tiles are smooth and 20% of the black tiles are smooth. Now when you put your hand in the bag and select a tile and it is smooth, what is the probability that it is a red tile given the new evidence?
Bayes theorem sets out a simple but powerful calculation to come up with the answer. The probability that the tile you are holding is red, given that it is a smooth tile, is equal to the probability of the tile being smooth, given that it is a red tile, times the probability of it being red divided by the total probability of the tile being smooth (i.e., red and smooth and black and smooth).
Or (0.8 x 0.7)/((0.7 x 0.8) + (0.3 x 0.2)), which is about 0.9 – in other words, a 90% chance that the tile you are holding is red.
This simple formula, although heavily criticised as allowing for too much subjectivity — how do you “know” or weight a prior probability? — has become a major pillar in the philosophy of science, statistics, and AI. It is also used in the betting industry, but I will come to that later.
It can be used to determine the probability that a person does have a specific cancer, given the underlying rate of this cancer in the population and the false-positive rate inherent in the test.
One of the most misunderstood areas where the theorem comes in handy is with the use of DNA evidence in a trial. Imagine a defendant at a murder trial and DNA has been found on the murder weapon, which matches the DNA of the defendant. A DNA expert says that the chance of a random person matching this DNA is one in a million. The prosecutor in summing up says, “There is only a one in a million chance that the defendant is innocent”!
The prosecutor’s statement is a misinterpretation of the DNA expert’s statement. What the jury needs to know is, “What is the probability that the defendant is guilty, given that the DNA matches?”
Bayes theorem can show how wrong the prosecutor was. Assume the population of the city is one million and the DNA test has false-match rate of one in one million. Then the calculation comes out to be about 50% rather than 99.9999%, provided there is no other evidence linking the defendant to the crime.
An easy way of looking at is, if all of the people in the city were tested, one person is guilty and they will provide a match to the DNA, and because of the false positive rate you would expect one additional false positive result. Hence, when confronted with a DNA match, it could be the killer or it could be the false positive, a 50% chance that the defendant is the killer.
Bayes is also used in search and rescue, helping Coast Guards, for example, to narrow down where to look for a drifting boat, given the tide, weather, and currents.
What has this to do with gambling? Everything!
Bayes can help with determining the odds of the outcome of a sporting event — what is the probability football team A will win against football team B, given that the team’s main goal scorer will not be playing due to an injury?
Bayes can calculate the probabilities of horses winning races given the competition, a horse being withdrawn from the race, the condition of the track, etc. Or the likelihood of a poker player bluffing, given that they have just placed a big bet.
Bayes is used in the algorithmic trading models of financial funds, estimating the probability of stock or other asset-class price movements, given new evidence of an earnings report, the movement of the price a related stock, or something similar.
Bayes was a devoted theologian, who wrote about the evidence for God’s existence and how to justify belief under uncertainty, which possibly led to his theory. I wonder how he would feel, if he were alive today and knew just how important his work had become in the field of gambling?
