Over on my quiz blog, I've taken a brief detour into the world of Eurovision, specifically the difficulty of predicting which countries will qualify from this week's semi-finals. The background to this post can be found there, whereas here I'll be going into slightly more detail about the maths behind the results. (And in the unlikely event you've arrived here first, I'd strongly recommend reading the original post first.)
Starting with the first semi-final, the problem is relatively straightforward to summarize: you simply need to pick 10 winners from a field of 16. (To aid intuition, it's helpful to notice that this is equivalent to picking 6 losers from the same field, but in either case the solution is relatively straightforward.)
As always it's a good idea to start simple. To get all 10 correct is a simple matter of choosing the right 10 countries from 16. There are 8,008 ways to choose 10 items from a list of 16 (see here if you're unfamiliar with how we can arrive at this number so effortlessly), so you have a 1 in 8,008 chance of picking all 10 qualifiers if you choose at random. Pretty slim, but not too ridiculous.
Next, say you got 9 correct, how good is that? To answer this we need to know the probability of picking at least 9 qualifiers or, equivalently, the probability of picking exactly 9 or exactly 10. We have the latter already, so what about the former?
We know there are 8,008 different ways to choose the 10 countries we think will go through. The question, then, is how many of these 8,008 ways correspond to having exactly 9 correct predictions. The answer comes fairly intuitively if we imagine trying to deliberately construct a set of 10 predictions made up of 9 winners and 1 loser: we just choose 9 of the 10 countries who qualified and 1 of the 6 who went out. There are 10 ways to choose 9 things from 10, and 6 ways to choose 1 from 6, giving us 10 x 6 = 60 ways to pick 10 countries of which exactly 9 will progress to the final. Adding this to the 1 way that gives us all 10 qulifiers means we're looking at 61 of the 8,008 possible ways to predict 10 countries: our
Eurovision-loving monkey has about a 1 in 131 chance of picking at least
9 qualifiers correctly.
This method easily extends to less successful attempts. For exactly 8 correct predictions we need to choose 8 of the 10 winners and 2 of the 6 losers. There are 45 ways to do the former and 15 to do the latter giving us 45 x 15 = 675 ways to get exactly 8 correct. Add this to the 60 ways to get exactly 9, and 1 way to get exactly 10, and we hit 736 out of 8,008, or about a 1 in 11 chance of getting at least 8 qualifiers correct. The method extends downwards for less successful predictions (and you can get the full results table over in the quiz blog post).
The only other question is how to modify this to work for tonight's 17-country semi-final. The same logic applies, and most of you could probably work this out yourself from here. Rather than 10 winners and 6 losers, you're now looking at 10 winners and 7 losers. There are 19,448 ways to pick your 10 qualifiers from the 17 countries, and just 1 of those will give you a full set: you already have less than half the chance to hit a perfect 10 than you did on Tuesday.
What about 9 correct? Applying the same procedure we can easily see that there are still 10 ways to choose 9 of the 10 winners, but there are now 7 ways to choose 1 country from the set of 7 losers. This gives 10 x 7 = 70 ways to get exactly 9 correct qualifiers in the 17-country set-up, or 71 out of 19,448 (1 in 274) ways to get at least 9 qualifiers right on the night. Again, full results can be found on the quiz blog.
That's all there is to it. A neat problem, I think, with a similarly neat solution. It's also an interesting lesson in how easily numbers can deceive: getting 7 out of 10 qualifiers right on Tuesday might have seemed good, but it's the same as identifying just 3 of the 6 losers. You'll have still done (slightly) better than a monkey, but it's probably not quite something to sing about.