Written by Colin+ in probability, sport.

A reader asks:

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.

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.

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.