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How we misjudge probabilities

Like the previous blog entry, this blog post was written by Joachim, our admin for the German-speaking BettingExpert.

Identifying value is an integral part of successful betting; anyone arguing that is probably losing in the long run. In order to identify value we need to have a good understanding of the probability of an event to happen - and we do need to understand what probability the bookmakers assume and whether or not they are wrong with that assessment. There's a bit of a problem though: The human mind has a track record of playing tricks on us in certain situations, and this also happens with probabilities. Quite a lot actually.

The Monte Hall problem

A famous example for this is the so-called Monte Hall problem. Consider the following scenario:

An unbiased game-show host has placed a car behind one of three doors. There is a goat behind each of the other doors. You have no prior knowledge that allows you to distinguish among the doors. 'First you point toward a door,' he says. 'Then I'll open one of the other doors to reveal a goat. After I've shown you the goat, you make your final choice whether to stick with your initial choice of doors, or to switch to the remaining door. You win whatever is behind the door.' You begin by pointing to door number 1. The host shows you that door number 3 has a goat.

Monte Hall Problem

Now, for the question: Do you think your probability of choosing the correct door increases by switching to door two, or do you believe it remains the same, even if you stick to your initial choice of the first door? Intuitively most of us are inclined to believe that it doesn't matter if we switch or not – we assume the probabilities are 50% for both options. But as it turns out, if you don't choose to switch, your probability is in fact only 33.3% ! - Accordingly, if you always switch, your probability of being right is a startling 66.7%, or two out of three times.

You're kidding me, right?

Actually, no. There is a lot of ways to represent the possible outcomes for this scenario, and in fact there is an excellent wikipedia article about the Monte Hall problem that discusses all possible explanations at length; but the most simple way of putting it is the following table, that covers all possible arrangements of this experiment. This particular table assumes you always pick Door 1 – but obviously this is applicable to every other door as well. So here we go:

Goat/Car Table

Okay, but why is that?

Critical to understanding this problem is that the TV host doesn't always have a choice (since he must not reveal the car). If behind the first door you pick there is a goat (which happens two out of three times), the TV host has only one other goat to show you. Thus if you switch you will be right 66.7% of the time. In essence, the TV host offers you additional information. If you choose to disregard that information and not switch, that means you're stuck with the same probabilities as if there had not been a TV host to show you any goats in the first place.

Another way to understand the Monte Hall problem more intuitively is to dramatically increase the number of doors involved. Imagine there are 1,000 doors, you pick one, and then the TV hosts opens 998 of the other ones. You are now left with two doors out of the original 1,000 – do you think it's better to switch, or would you rather stick with your first choice?

The birthday paradox

The birthday paradox is another great example of how we tend to misjudge probabilities significantly at times. The birthday problem, as it is also called, refers to the likelihood of any of the people in a given group having their birthday on the same day. How big do you think that group would have to be for that probability to reach 50% and 99% respectively?

The correct answers are just as amazing as for the Monte Hall problem: To have the probability of at least two people in a given group (none of them twins) having their birthday at the same day reach 50%, all you need is 23 people in that group. To reach 99%, all you need is 57 people. Talk about counter-intuitive.

Birthday paradox picture

Again, why is that?

Basically, we are guessing numbers that are a lot higher than the actual answer because we tend to make the wrong assumptions. It's very important to remember that we are looking for the chance of any two people of the group having their birthday on the same day. If we take one specific person with whom the birthday of another member of the group has to match and have 23 members in the group, there are only 22 chances for a matching birthday. If you are looking for the probability of any two members of the group having their birthday on the same day, you actually have to look at 253 pairs overall (23 times eleven pairs), which makes the actual probabilities easier to understand. If you want to go into this deeper, the detailed wikipedia article about the birthday problem is also excellent, but does require a bit of a statistical background.

The Hole-In-One Gang

It is important to note that misjudging probabilities is not necessarily a problem for punters only – in fact it can affect the bookies just as much, and lead to sometimes exceptional value bets. A famous historical example is the Hole-In-One Gang, that consisted of two very crafty and sharp punters called Paul Simmons and John Carter.

Hole in one

Back in 1991 they calculated the probability of any given golfer in a tournament hitting a hole-in-one. As it turns out this isn't as unlikely as we all think – in fact that probability hovers around the 50% mark (for any given tournament). They traveled around the UK placing as many bets as they could (remember, this is the pre-internet era), as bookmakers all over the country were happy to give them huge odds on this bet, with odds ranging anywhere between 4.00 and 101.00 in decimal odds – in short, exceptional value. Obviously, most of these bookmakers were too lazy to do the necessary stats checking – but also note the obvious similarity to the birthday paradox. Many of the bookies involved were apparently relying on their intuition to quote the odds – and just like in the birthday problem made the mistake to confuse the odds of one specific player to hit a hole-in-one with that of any player in the tournament achieving a hole-in-one. It's what happens if you don't do the math.

As it turned out, hole-in-ones were scored in three of the four major golf tournaments that year, and Simmons and Carter obviously cleaned up big time – they were reported to have made at least half a million pounds sterling in profits. Back in 1991 that was a lot of money.

So now what?

I think there are quite a few valuable lessons to learn from all of this. For one, human intuition can play cruel tricks on us. But what is more, that isn't necessarily something that hurts our betting:
When we play our cards right (or bets for that matter), it's also a phenomenon that can help us beat the bookie – or whoever the poor guy is on the other end of our bets on Betfair.