Probability defines the world we live in. It should drive our decision making process. Unfortunately, most of us lack an understanding of probability. We often rely on pure guesswork rather than calculated risks when making decisions. We let cognitive biases influence our decisions. We constantly overestimate and to a lesser extent underestimate the likelihood of an event. In its simplest terms probability is the likelihood of an event occurring. This article hopes in improve the readers understanding of probability. To that effect, I will try to keep things as simple as possible.
Smart people make stupid decisions. This is especially true when they gamble. This article is not an opponent nor proponent of the betting industry. However, there is a focus on bookmaking and gambling as this is an area where people can put what they learn into action effectively. People are always going to gamble, especially foolishly. However, they should at least be as informed as possible when they do decide to make a bet.
The much greater than a million dollar question
To demonstrate how poor we are at calculating probability, I would like you to ponder a simple question; how many ways are there to rearrange a deck of cards? The title of the section might be a giveaway that the answer is much higher than most would have thought. However, I would be quite surprised for someone to instinctively guess the correct answer……
There are 80,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
different ways to shuffle a deck of cards.
That is an 8 followed by 67 zeros - eighty unvigintillion. Let all those zeros sink in. This is a ludicrously large number. It is unfathomably large, inconceivably large. Read this next sentence twice. There are more ways to arrange a deck of cards than atoms on earth! There is a seemingly infinite (seemingly!) number of ways to shuffle a deck of cards. Consider how many card games must have taken place across the world since the beginning of humankind. No one has held or will likely ever hold the exact same arrangement of 52 cards as you did during that game. That is, if you shuffle a deck of cards, in all likelihood, no one has ever held a deck of cards in the same combination. A shuffled deck of cards is an (almost) truly unique object.
How can there be so many different ways? Well, the mathematics behind it is rather simple. It is known as a factorial, or in this case 52 factorial, denoted as 52! You multiply each number by the next number in descending order until you reach 1. That is, in this case 52 x 51 x 50 x 49 x ……. x 3 x 2 x 1.
This calculation will lead to the ridiculously large number above. It is quite a simple concept if you give pause to think about it. Imagine a deck of cards face down. The first card you draw from the deck of cards can be any of the 52 cards. For the next card, you know it cannot be the first card you have drawn but it can be any of the remaining 51 cards. Similarly, once you have drawn this card you will know that the next card drawn can be any of the remaining 50 and so on. You continue this process until you have only one card remaining. By process of elimination, you will know what the final card is. To get the total possible combinations you simply multiply the number of possible cards for each card you draw from the deck. That is, 52 possible cards for the first card, 51 possible cards for the second and so on. This leads to the calculation known as a factorial.
It is quite common for us to underestimate the various number of combinations of events that can occur. If we underestimate the number of events that can occur, we overestimate the probability of the event occurring. By over-estimating probability we will make ill-informed, poor decisions. This can be particularly true when we gamble.
Bookmaker’s odds & implied probability
What is the implied probability of a 2/1 bet? That is, ignoring the profit margin bookmakers will seek to exploit, what is the likelihood of an event with 2/1 odds coming true? Many people would answer this question confidently, stating a ½ or 0.50. These people would be wrong. Take a moment to think about. If you bet €10 on a 2/1 bet you stand to make a profit of €20 euro, for an overall return of €30. If the probability of the event being a success were ½ and for each bet you made you stood to make a return 3 times the size of the stake, the bookmaker would stand to lose a lot of money fast. In reality, the implied odds are 1/3 or 0.3333.
There is a simple formula to follow when trying to calculate the implied odds. For a bet which has odds of A/B, the implied probability is B/(A+B). That is a horse with odds of 3/1 has an implied probability of winning of 1/4 or 0.25 and so on. You should think about the implied probability when deciding whether or not to make the bet. In addition, it might be useful to think of a horse priced at 2/1 like this, if this exact same race occurred 3 times, would you expect the horse to win at least once? If not, it might not be wise to make the bet.
It is important to note, that there are many factors that influence the odds offered by bookmakers. Among other things, they must leverage against other bets made by other customers, a large bet in a small market can seriously affect the odds on offer. However, in simple terms, the real probability of the event coming true will be less and this is the margin the bookmakers hope to exploit to turn a profit. It is important to remember when gambling that the betting industry’s goal (as it should be) is to make a profit and you are their means to make said profit. A great example of this margin that the house hopes to exploit is in a standard roulette wheel. The numbers on a roulette wheel range from 0–36. If you were to put a punt of €1 on 17 black, you would be offered odds of 35/1, for an implied probability of roughly 0.0277. However, due to the inclusion of the green zero the actual probability of the tiny white ball landing on 17 black is 0.027027. Similarly, the odds of it landing on black or red is actually roughly 0.4865 rather than 0.50. This tiny margin has made vast, vast, vast sums of money for casinos.
It must be noted, that the odds presented here are odds that have been traditionally used by bookmakers in Ireland. However, presenting the odds in decimal format is increasingly more popular. Decimal odds demonstrate the overall return. For example, decimal odds of 2.0, means for every €1 you bet you get €2 back and is equivalent to evens (1/1) in traditional odds. Decimal odds of 4.5, means for or every €1 you bet you get €4.50 back and so on. It is quite simple to get the implied probability from decimal odds. If the decimal odds are A the implied probability is 1/A. Furthermore, there are American style odds which is a bit more convoluted, you can find it explained quite well here.
Most people who have ever made a bet on a horse race would be familiar with terms such as ‘double’, ‘triple’ or ‘accumulator’. A double would be you bet on one horse to win one particular race and another horse to win another. A triple would be 3 horses to win 3 separate races. The term accumulator is used as the number of horses and races increase - a quadruple or quintuple just do not have the same ring to them as a double or a triple. These races are assumed to be independent. That means that the events have no influence on one or other. Horse A winning race A has (all things considered fair) no effect on the likelihood of horse B winning race B and neither should have any effect on horse C winning race C. Thus it is easy to calculate the likelihood of this event occurring. The probability of all horses winning would simply be Probability(A) * Probability(B) * Probability(C). This makes it very easy for bookies to calculate to returns for a bet like this.
To explain this concept a little further lets propose a bookmaker offers odds of evens (1/1) for horse A winning race A, 3/1 for horse B winning race B and an outside horse C with odds of 9/1 for race C. From the previous section we know that the implied probabilities for each are 0.50, 0.25 and 0.10. Thus the probability of all 3 horses winning (as each race in independent) is 0.50*0.25*0.10 = 0.0125 or 1/80. The odds on offer for this should be, reversing the formula provided earlier, 79/1. Another way to think about it is that in each race you are just raising the stake by the previous race wins. This will still work with races happening simultaneously. That is €1 at evens returns €2, €2 at 3/1 returns €8 and €8 at 9/1 returns €80, which is equivalent to betting €1 at 79/1. However, as good as it is to understand how these returns are calculated, it really needs to be understood how (un)likely it is for a bet like this to pay off. Understanding this (un)likelihood will help you make more informed decisions.
However, these simple calculations do not apply to dependent events. For example, the probability of Paul Pogba being first goal scorer and Man United winning a match are not independent. If Paul Pogba scores first it dramatically increases the chances of Manchester United winning - although as of late one would still think this to be quite improbable. If we were to calculate the combined odds for both as previous, we would be miscalculating the odds completely. Not to get bogged down in mathematics but this is a much more complex calculation when events are not independent. There would have been a time when bookmakers would not offer prices on dependent events such as this. However, increases in computing power and greater reliance on statistical models, rather than human judgement, have enabled bookmakers to offer accurate prices on more complex bets like this. The Irish company Banach Technology are leaders in offering solutions to the more complex and interlinked bets that are in increasingly higher demand by punters.
Bias plays a huge role in our mind’s primitive calculation of probability. Humans have a tendency to see patterns everywhere. We often find ‘patterns’ that are merely a coincidence. For example, you flip a (fair) coin 6 times and it lands on heads each time, what is the likelihood that it will land on heads if you flip the coin again for a 7th time? If you believed that the chances of it being head would be higher due the previous flips, you would be wrong but not alone. The chances (given that it is a fair coin) of landing on heads are 50%, such were all the other previous coin flips. Each coin flip is independent. The tendency to think that future probabilities are altered by past events, when in reality they are unchanged is known as the Gambler’s Fallacy.
This idea that you are on a streak can be dangerous when gambling. Take for example roulette. Each spin is independent. Past spins have no influence on the next spin. You have won the last 3 times and feel indestructible but the the previous spins have zero influence on the next spin. It is often smart to walk away when you have (against the odds) made a profit. Furthermore, the Gambler’s Fallacy can also work detrimentally in the reverse. People continuously betting and losing thinking ‘this next time I will have to win after all those losses’. The idea that if you have lost the last 5 spins you must win the next. Above all else, it can be a deadly and deathly issue if you are playing Russian roulette….
There are an array of various cognitive biases that play a part in how we naturally and fundamentally incorrectly perceive the world. Another example is anchoring. This is when an individual relies too heavily on an initial piece of information offered i.e. the “anchor” when making decisions. There are obvious ways how this could negatively affect the decision making process. SIG, a quantitative trading firm who base their trading strategy on poker* strategy and quick accurate probability calculation, outline how anchoring (and other cognitive biases) effect the decision making process.
These are just two examples of the wide range of cognitive biases we fall prey to often, without realising. In my opinion they are very thought provoking. If these biases interest you, you can check out this more comprehensive list. On a side note, Israeli psychologists Daniel Kahneman and Amos Tversky did amazing and ground-breaking work in the study of judgement and decision making. The former was awarded the Nobel Prize in Economics for their joint work (the accolade is not awarded posthumously). Their work was revolutionary in the field we now know as behavioural economics. I would highly recommend the book “The Undoing Project” by Michael Lewis (most of his work is phenomenal) about the pair’s friendship and work. For work written directly by the authors, checkout “Thinking, Fast and Slow” by Daniel Kahneman.
There are many factors that we are not consciously aware of that play decisive roles in the decision making process. As a result of this, we often make irrational decisions. By using probability in our decision making, especially when gambling, we can make much more calculated and rewarding choices. Probability is extremely vast and a profoundly interesting topic, for me at least. Probability calculations can get extraordinarily complex. However, even the most rudimentary of understanding can go a long way. This is the same for Mathematics as a whole. I would encourage everyone to try and move past any PTSD they have from Maths from school and dip their toes into the world of Mathematics. Hopefully, some of you will dive head first!
*Note: It might have been noticed that there is essentially no mention of Poker in this article. Poker probability and decision making merits at least an entire article. There is so much that can be discussed it would be a disservice to try and fit it in here.