# Probability and prediction

Is it easier to predict every game in the first round of a tournament1, or to pick the eventual winner?

There's a competition run by Sky, which is free to enter; you win £250,000 if you correctly predict the scores in six Premier League games. It's free to enter. A quarter of a million quid, if you get six scores right, for no stake.

Mr Murdoch, for all his flaws, isn't a mug. Your chances of correctly predicting a scoreline are - at best - about one in eight. Predicting six? One in $8^6$, which is... one in a quarter of a million. (Which suggests that getting you onto the website is worth about £1 to Sky, but that's a different story for a different day.)

Why do I bring that up? Because predicting several things correctly is REALLY DIFFICULT, even if you're an expert2 In general, the more things you have to predict, the more difficult it is.

### An evenly-matched contest

Imagine a 64-team contest where all of the teams are perfectly evenly matched. (Maybe they're playing a game that doesn't involve any skill.) The probability of you predicting all 32 games in the first round correctly is $0.5^{32} = \frac{1}{2^{32}}$, or one in about 4.3 billion. I'll say that again: one in 4.3 billion.

Meanwhile, since all of the teams are equally likely to win, you have a one-in-64 chance of predicting the winner: it's clear that predicting the winner (who has to win a handful of matches) is much easier than predicting the outcomes of all 32 first-round ties.

### Something more realistic

Obviously, real tournaments aren't 50-50 all the time. You'd expect Novak Djokovic to beat me at tennis (or just about anything else other than maybe mental arithmetic) pretty much all the time. In a tennis tournament, most games have a very clear favourite, which makes things a bit easier.

So I looked up the odds for the US Open tennis tournament, which - at the time of writing - has just finished. Multiplying the odds of the second-round3 winners together, I got a probability of about one in 100,000 - ten of the games had a very hot favourite expected to win more than nine times out of ten, which makes it easier to pick than the fair-coin version.

There was a surprise winner in this tournament - the Croatian Marin Čilić winning his first Grand Slam title, beating another surprise finalist, Kei Nishikori of Japan. Čilić, before the tournament was - coincidentally - about a 65/1 shot, similar odds to the coin-toss champion from earlier on.

Again, the chances of picking even a surprise winner (Djokovic was considered in some places something like a 4-in-10 shot) are much, much higher than predicting a large number of independent events.

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

1. For the sake of argument: a 64-team tournament []
2. Accumulators, where you bet on several results to go the right way, are mug bets, which is why the bookies promote them. []
3. the round of 64, to keep things fair []

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