Expected Value and the large of large numbers.
Both Expected value and the law of large numbers are your friend — when the conditions are correct.
In probability theorem, the law of large numbers described the expected result of performing the same experiment a large number of times. When analysed in betting and strategy performance it will provide an indication of how well your model or strategy is working.
In any sports market, we have the closing price. A strong proxy for the true odds but it is in fact highly plausible that we will never know the true odds of a team winning or losing. The closing odds in comparison to the opening odds reflect all the information known to the market at the time of play. Once the game kicks off thousands of hidden variables are introduced — a red card, a missed penalty, poor or good team performace or injuries to name a few.
Bernoulli, (either Jacob or James as sources contradict who actually founded this theorem ) experimented with the classic coin toss scenario using his Binomial Distribution. The binomial distribution models the number of “successes” that occur when conducting a series of binary experiments (often referred to as “success” and “failure”) with a known probability of success.
The key characteristics of a binomial distribution are as follows:
- Fixed Number of Trials: The binomial distribution considers a fixed number of trials or experiments, denoted as “n”. For example, flipping a coin 10 times or conducting 20 customer surveys.
- Independent Trials: Each trial or experiment is assumed to be independent, meaning that the outcome of one trial does not affect the outcomes of other trials.
- Two Possible Outcomes: Each trial has only two possible outcomes, often referred to as “success” and “failure.” These outcomes are mutually exclusive and exhaustive.
- Constant Probability of Success: The probability of success, denoted as “p,” remains constant across all trials. The probability of failure is given by 1 — p.
He was concerned with the number of times heads was occuring and noticed that as the number of coin flips he did and jotted down the standard deviation increased too. The standard deviation is a measure that quantifies the amount of variability in the data, ie howspread out the values are from the mean or expected value. For example, with ten experiments of a coin toss, n = 10, p=0.5 and the std = 1.58 tosses. When he ran 1000 coin flips the standard deviation increased ten fold.
This was not a coincidence, as n increased 100 fold which is 10 squared. The next two plots I will show is the normal distribution of ten and a thousand tosses of a coin. The mean is 5 heads and the chances of seeing 1 head out of 10 tosses is about 1%, in 10 results there is a bigger chance of witnesses improbable results than there is in a larger experiment size.
When the number of tosses rises to 1000, the chances of seeing 100 heads (the same 1 in 10 proporation) is extremely low, much lower than the 1% for the ten coin tosses.
How and why does it concern you as a bettor? The law of large numbers is one of the most important probability theories there is when gambling since the difference between winning or losing long term is defined by it. Gambling should never be thought of as a short term plan nor should you judge a strategy over a small sample or time frame. You can consider that it is a sort of time series analysis that is based over longer term yields. In the short term luck is more prevalent to result in losing runs, if the strategy is sound.
This can be summarised as — If an expected value strategy that beats the margin implied by bookmakers is sound the more bets that a bettor will place obtaining the expected value (aim for 5% as anything more than that is quite an achievement) the higher likelihood that the results will move towards the expected value. For the coin toss example, we seen that over a larger sample the standard deviation of seeing outliers decreased significantly as the trials increased. To that end, the longer a bettor goes placing negative expected value bets it is inevitable that they will lose money, anyone can get lucky in ten bets placed with margin implied, but over a sample of a 1000 it is extremely unlikely to be profitable.
How long does it take for the law of large numbers to kick in? Is it a 1000 bets? Fortunately, its not that high. The speed at which the expected value kicks in is decribed by the power law taking the easily explained example of the fair coin toss.
The power law, where the quantity of the standard error (how spread out the values are from the mean) changes as a function of the power of another variable the coin toss probability (but when your concerned with betting it would be the number of bets)
The power in the fair coin toss scenario is -0.5 which means that as the number of coin tosses increase by just 4, the size of the standard deviation decreases by the square root of 4. It halves. This means that the standard error can drop by 0.50 in ten coin tosses to under 0.10 in roughly 100 coin tosses.
Lets assume, your model has an expected value of 11%, and that you place all bets at 2.0 (even money) where the true price is 1.8 (or 4/5). The formula for calculating the expected value is:
Expected Value = (BookmakerOdds/TrueOdds — 1) = (2.0/1.8–1) = 0.11% meaning that over a long sample you should expect to make 11% as long as bets are singles.
If your working on a model, please take the George Box mantra to heart: “All models are wrong, but some are useful”. Validate your strategy by back testing, or placing dummy bets, record these bets in a spreadsheet and keep track of the numbers. If your too excited to hold off, keep stakes small to start off with. If your model is wrong it can be a costly financial mistake to make.
The topic of this article was inspired by Joeseph Buchdahls book “Monte Carlo Simulations for the aspiring sports bettor”. I would reccomend all his books for anyone interested in sports betting and remember…..
“Never place a negative expected margin bet”, but when your drunk thats perfectly acceptable.
