Luck or Skill

To be a successful sports punter over the long term requires skill sufficient to find an edge over the bookmaker’s odds. Luck can be profitable in the short term, but unless we have that edge, luck will eventually run out; it’s not a question of if but when. Once we have a profitable betting history, the next step is to determine whether it has happened by good fortune or genuine forecasting ability; or more specifically what is the probability that it has arisen by chance. Of course there are those punters who still believe that a winner by definition is evidence of skill. Such punters will be one or more of the following: ignorant; in denial; bankrupt. By contrast, those with even a basic understanding of probability will know that with so many random influences operating in a sporting contest, it is simply impossible to rule out the possibility of good fortune when a solitary bet wins, or conversely bad luck if it should lose. To begin to separate the influences of luck and skill, we need to analyses a much longer series of bets; just how long will form much of the scope of this lesson. Short of actually learning how to find an edge, this is perhaps the most important lesson there is to learn in sports betting.

Consider flipping a coin 20 times. The chances of returning x number of heads are governed by a discrete probability distribution known as the binomial distribution (see below). If we know that there is a 50% chance of flipping heads and a 50% chance of flipping tails on every coin toss it’s a simple enough exercise to do the maths and find those probabilities of each possible outcome. [A software package like Excel offers easy calculation of binomial probabilities using the function BINOMDIST.]

Predictably, since heads and tails are equally likely to occur on each toss, the most probable number of head and tails after 20 coin tosses will be 10 and 10 respectively. But this does not mean that we will see 10 and 10 all the time. In fact, in this example, returning exactly 10 head and 10 tails has an only an 18% chance of occurring. Sometimes we might see exactly 9 heads and 11 tails (16% probability), or 12 heads and 8 tails (12% probability), or very occasionally 5 heads and 15 tails (1% probability). What such a probably distribution does show, however, is that for the majority of occasions the number of heads will fall within a fairly narrow band concentrated around the average. For example, in nearly three-quarters of occasions, the number of heads will fall within two throws of the average of 10, that is to say 8, 9, 10, 11 or 12. Such a distribution also allows us to calculate how likely certain outcomes are of occurring. For example, from this probability distribution, we know that the chance of tossing at least 14 heads from 20 coin tosses is roughly 6% (the cumulative probabilities of tossing 14, 15, 16 17, 18 19 or 20 heads).

Whilst sporting outcomes are not completely arbitrary events like coin flipping, they do show a significant degree of inherent randomness. And just like coin flipping, the frequency of wins and losses will be distributed in a similar manner. If the odds are fair, an extended period of blind (unskilled) betting would expect, on average, to break even. The chances of exactly breaking even would be fairly small (as is the chance of landing exactly 10 heads in 20 coin tosses), but the probability of being in a narrow range of small loss or profit about the average of zero would be high. Alternatively, where the odds are unfair, we would expect there to be a high chance of losing an amount on turnover that closely reflected the bookmaker’s overround.

So what has all this got to do with separating luck from skill? Well, suppose from 20 coin tosses we landed 16 heads. According to binomial theory, we should expect to see 16 or more heads form 20 coin tosses just 6 times in a 1,000, assuming that the coin is unbiased. Whilst it’s possible this could happen by chance we might start to wonder whether the coin was perhaps biased. Let’s suppose we throw the coin a further 20 times, and this time see 17 heads. That’s a total of 33 heads in 40 coin tosses. Probability theory tells us the chance of see a least 33 heads is just 1 in 47,307. Now we would be pretty certain that something was wrong with the coin. And if we tossed it a further 20 times and saw a further 18 heads for a total of 51 in 60, the odds of that are a mere 64,829,077/1. True, there will always remain a finite (non-zero) probability that this could happen by chance (remember people win national lotteries at odds of millions to 1 purely by chance), but the far more likely explanation is that the coin is weighted in favour of heads.

We can perform exactly the same statistical analysis for sports betting results. Provided that every bet we placed was for fixed win expectancy as for coin tossing (where it is 50% for either heads or tails) we could perform exactly the same binomial analysis. For example, suppose we had a particular liking for 3/1 shots (4.00 in European decimal format), where the probability of a win is 25%. Suppose we placed 100 bets and had 50 winners. Binomial theory allows us to calculate that the probability of seeing at least this many winners (in other words 50 or more) purely by chance is roughly 1 in 15 million.  We’d be damn near as certain as we possibly could be that something else other than chance was accounting for this; the most obvious explanation, of course, would be skill.

Sports punters, however, rarely choose to bet the same odds, and for those who favour Asian handicaps and American spreads betting ties also become a possibility. Binomial is then not really an option for statistically testing a betting history. Instead, it is easier to analyse the actual betting returns themselves rather than the number of wins and losses, and compare them to what we might otherwise have expected, on average, to achieve. The best statistical test of this purpose is what is called the t-test.

Whilst the mathematics that lay behind the t-test are a little different to the binomial, the important point to grasp is that it is essentially performing exactly the same task; that is identifying how likely a series of betting returns could arise purely by chance. To achieve this, the t-test essentially compares one average to another, and looks to see if there is a significant difference between the two. For sports betting histories, this means comparing our actual average profit per bet to a theoretical profit per bet that would be expected, on average, to occur if only chance was at work. The former is straightforward to calculate: it’s simply our decimal yield or profit over turnover. For example, a 10% yield (decimal 0.1) is equivalent to making an average profit of 0.1 units for every 1 unit we stake. The latter is essentially the profit expectancy based on the bookmaker’s odds. Recall from the lesson on singles and accumulators that profit expectancy = (1/decimal overround) – 1. So if the typical overround for the things we were betting on was 108% (1.08) then our average profit expectancy would be -0.074. Our t-test would then compare our average of 0.1 to the expected average (that is what our bookmaker would predict we would achieve) of -0.074 for our sample of bets.

Of course, punters rarely bet with just one bookmaker, but instead use a number of them with the view of obtaining the best price available for each particular bet. We’ll look at this in more detail in the lesson on odds comparisons. For now it is sufficient to say that for the majority of bets, by taking the best price available across a market of bookmakers, we can reduce the effective overround or natural disadvantage we face virtually to zero. That is to say, provided we are diligent in finding best market prices, we shouldn’t actually do worse than roughly breaking even over the long term. Consequently, our average profit expectancy is now about 0.00 and it is to this figure that we would compare our actual decimal yield.

At Tipster Bay, we’ve provided a useful probability calculator, based on the mathematics of the t-test, which for most betting histories will estimate the probability that it will have arisen by chance. For those histories where stake sizes do not vary too much from level staking, this probability is a function of the number of bets in the history, the betting yield, and the average betting odds. If your betting history is not to level stakes, make a new one based on your actual bets and odds and assume that it was. Find out what your yield to level stakes was – remember that your level stakes yield is probably the most reliable indicator of your betting edge – and then plug in the numbers. Try out other betting histories with different odds and different numbers of bets. Longer profitable records (that is to say histories with more bets, not more time, since time in this context is irrelevant) are less likely to have arisen by chance, all other things (yield and average betting odd) being equal, just as we saw for our coin tossing experiment. Profitable records with shorter average betting odds are less likely to have occurred by chance, again all other things (yield and number of bets) being equal, simply because there is less inherent randomness at work with individual bets of greater win expectancy. For example a yield of 10% from 250 bets with average odds 3.00 has roughly a 14% probability of arising by chance (assuming that the expected average we are comparing it to is break even). By contrast, that figure is less than 1% for a similar record but with average odds of 1.50. And predictably, of course, more profitable records (for equivalent betting odds and history size) will be less likely to have arisen by chance.

Typically for statistical testing in the sciences we assume that if the probability of something happening by chance is less than 1%, we begin to attribute some real significance to this result. Anything more and it’s simply too large to rule out chance as an influencing factor. Anything smaller and we begin to take note. Of course, this choice of 1% is purely arbitrary, not based on the statistical testing itself, but on the point of views of the majority of the mathematics community.

Estimating the probability that a betting history has arisen by chance is all very well and good, but it still doesn’t actually tell us whether it is based on skill. Well, unfortunately, no statistical test can do that, since there is always that finite and no zero probability that chance has been a factor. When we use things like the binomial distribution and t-test we must always keep in mind what their results are telling us. When we have found that a particular betting history has less than a 1% probability of having arisen by chance, that is not the same thing as saying there is more than a 99% probability that it has arisen by skill. Such statistical testing can say nothing about what is causing the betting profits to accumulate, since it is not designed to make any comments about causality. It is up to the user of such techniques to make rational but subjective judgements about what might be going on if the effects of chance are probably very small.  Naturally the most obvious alternative is predictive skill of the punter, but a t-test result can never tell us this on its own.

A key question we set out to answer was just how many bets does it take to find out if we have a genuine edge over the bookmaker, one that is based on skill and not luck. This lesson should hopefully have provided some ideas about how to go about doing that and how to interpret, and not misinterpret, any statistical analysis we might perform to this end. Those of you, who have had a go at analysing your own betting history, or that of a tipster, might be surprised at just how long it can take before we can start believing in any real ability. The table below provides a summary of the number of bets we would need to place for some odds/yield combinations before the probability that the profitability has arisen by chance alone falls below 1%. For some of the weaker performances, or histories with longer betting odds, it should definitely provide some valuable food for thought.

Next time someone tells you their tips are the best thing since sliced bread you’ll know how to challenge him to prove it.