Chances are probability isn’t your strong suit

First published in The National (Abu Dhabi) 28 Feb 2016

Who is going to come out winners today ? Will it be Manchester United or Arsenal,  Matt Damon or Leonardo DiCaprio ? These are the big questions for football fans and film buffs, vying to predict the outcome of today’s big Premiership game or the winner of tonight’s Oscar for Best Actor.

Success brings bragging rights and a reputation for being smarter than the next guy. But such kudos is not easily won, and those wanting the best chance of being proved right must wrestle with that most paradoxical branch of mathematics: probability.

For many of us, that word triggers memories of dull school lessons spent working out the chances of, say, picking a particular colour ball from a jar. Worse, even the simplest problems seemed to defy commonsense. After tossing a coin four times and getting heads every time, it seems obvious the chances of getting tails next time have increased. Except they haven’t: the textbooks insist the chances remain 50:50, because the coin has no memory of what happened last time, making each toss independent. Ok, but as we toss the coin more often, surely we’ll end up with roughly equal numbers of heads and tails ? Wrong again:  in fact, the numbers of heads and tails diverge over time.  What does “average out” is their relative proportions of the total number of tosses – in other words, their probability.

If you’re starting to feel baffled at this point, take heart.  Even the mathematicians who pioneered probability theory around 350 years ago made basic mistakes.  One of the greatest theoreticians of the Enlightenment, Jean-Baptiste d’Alembert, went to his grave thinking a coin is more likely to come up tails after a run of heads. But there’s a flip-side to this contrariness: probability routinely throws up surprises.

Take today’s game between Manchester United and Arsenal.  Predicting the outcome is far from easy, but one thing can be predicted with almost 90 per cent certainty: that at least two players on the pitch will have birthdays within a day of each other. This sound ridiculous; with just 22 players but 365 days available for birthdays, surely the chances must be less that 1 in 10. But this is where the normally irksome contrariness of probability makes things more interesting.

While there are only 22 players, there are over 230 ways of forming pairs from them and comparing their birthdays. That gives plenty of scope for finding a match. The chances are boosted even more by the fact that we’re not looking for a precise match, but for just birthdays within a day of each other. That makes finding a match easier, just as it’s easier to hit one of three adjacent targets than hitting just the one in the middle.

Actually calculating the chances of getting the birthday coincidence is pretty tricky, but the upshot is simple enough: 9 out of 10 football matches will have two or more players with birthdays within a day of each other. Probability theory predicts other coincidences at a football match. For example, it shows that around half of matches will have at least two players with exactly the same birthdays.  The chances of this coincidence are lower, but that’s because it’s more specific, demanding a precise match.

Don’t believe any of this ? Then check the birthdays of the players in, say, a dozen games between any two teams you like. You’ll find around half of the matches will feature exact matches, and most will have players with adjacent birthdays.

As so often with probability, however, there’s a bigger lesson here. We’ve all experience strange coincidences: bizarre encounters with old friends, dreams that seem to presage the future, eerie confluences of events. We find them unsettling because we think they’re highly improbable – and thus cannot be mere chance. Yet probability theory shows that the less demanding we are about what constitutes a coincidence, the more likely we are to encounter it.  And with “spooky” coincidences, we don’t make any demands at all: we decide what constitutes such an event after  we’ve seen it. That’s like claiming a brilliant goal after shooting in any direction and then moving the goalposts there after the kick.

Probability can do much more than merely protect us from silly beliefs, however.  Its laws also cast light on far tougher challenges – such as which team will win today’s Premiership clash. Contrary  to the impression left by those dull maths lessons, probability applies to more than just coin-tosses and balls in jars. That’s just one variety of it, called aleatory probability  (from the Latin for “dice-player”), whose focus is problems involving events whose outcomes are uncertain because they’re the result of randomness.

There’s another, much more interesting type, however: epistemic probability. From the Greek for “knowledge”, this deals with uncertainty caused by a lack of information. While we know that in the long run a coin will land on heads in 50 per cent of all tosses, many events with uncertain outcomes don’t even offer us that amount of insight.

Take today’s game: there are two teams, but it would be crazy to treat them like a coin-toss and assume they have an equal chance of winning. For a start, Manchester United is playing at home, giving them a well-established advantage. On the other hand, Arsenal are above them in the Premiership, and have won more of their recent fixtures. But how do we combine all this fragmentary knowledge ? Mathematicians have shown that the best way is via probability theory, which provides rules for combining what we know to update our beliefs about outcomes.

Of course there are easier ways of trying to predict outcomes, like gut instinct or guessing.  And not even probability theory guarantees to gives the right answer every time. But what is certain is that its rules give us our best shot of making sense of our uncertain world.

(Note added: Manchester Utd beat Arsenal 3-2)

Robert Matthews is Visiting Professor of Science at Aston University, Birmingham. His new book “Chancing It: The Laws of Chance and what they mean for you” is published today on Amazon

Leave a Reply