In the game show Who wants to be a Millionaire the most effective life-line is ‘Ask the audience’, which delivers the correct answer more than 90% of the time. Statistically it’s far more likely to be right than phoning a friend, even though, presumably, everyone picks the cleverest friend they know to phone in their moment of need.

This, I suppose, is at the heart of the Brexit situation.  Experts say that bad things are currently happening and that worse things are coming, but the public are sick of experts, the wisdom of the crowds has chosen for the UK to leave the EU and nobody wants to call the voters stupid for voting as they did.

Let’s take a step back.  Why does asking the audience deliver such impressive results?  There are, broadly speaking, two choices…

1. The audience is composed of geniuses
2. The audience are normal people, and some other force is at work

Let’s assume the answer is (2) and see what this other force could be.

Imagine a question is asked of such a difficulty level that only 10% of the general population know the answer.  If our audience has 200 members that then suggests that 20 of them know the right answer.

What about the rest of the audience? They don’t know the answer, but they’ve got a 1-in-4 change of getting it right, so they might as well guess.  If the guessing was totally random then we’d expect each of the 4 possible answers to get ¼ of the vote.

From our audience of 200 we have 20 people who know the answer and 180 that are guessing, so each of the four options should get 45 votes (¼ of 180), and the correct option will get those 45 votes plus the 20 votes of the people who know the correct answer.  So, with perfectly random distribution, three of the answers will get 22½% of the vote and one answer – the correct one – will get 32½%.

Democracy works!

Why then is the Ask the Audiences’ success rate high, but not perfect?

Consider what happens if we make our question harder, so that only 2% of the audience know the answer.

Now, from the 200, we have 4 who absolutely know the right answer and 196 who are guessing.  With perfect distribution we’ll get 3 answers that 49 people plump for and one answer, again the correct one, that 53 people select.

Democracy is still working, right?

Well, the killer here is the phrase “perfect distribution”, which means that everybody who doesn’t know the answer makes a completely random choice and that those random choices balance out so that everybody making a random choice spreads it evenly over the possible answers.

However, consider this question:

Which of these celestial objects is the smallest?

A. Mercury

B. The Moon

C. The Sun

D. Pluto

Now distribution isn’t perfect, because most people know that the sun is huge.  Some people don’t know that and some people will just select any stupid answer you give them (believe me, I work in market research), so a small minority will pick option C, but the rest will scatter their answers over the remaining 3 options.

Before we ask the audience let’s use another life-line and go 50:50.

Which of these celestial objects is the smallest?

A. Mercury

B. The Moon

C. The Sun

D. Pluto

We know that 4 people in our audience know the answer, the other 196 will vote based on what they think, using a whole raft of rationales to justify their answer.  The wrong option will get, statistically, 98 votes and the  right one will get 102.

In other words, even if everything goes perfectly, the final result will be 49% vs 51%. IF, and only if, it just so happens that all of those people guessing guessed perfectly randomly AND those random guesses happened to be distributed evenly.  If either of those assumptions is wrong, even by a little bit, the margin between right and wrong is so small that we end up in one of the situations where Ask the Audience delivers an incorrect answer.

If this was for the £1 million question would you gamble based on the 51% result and risk losing £468,000, or would you take the half-million and walk away?

If you take the half-million quid you’re not insulting the audience, just accepting that the majority of people who answered were guessing the answer.  The audience wasn’t filled with expert astronomers, it was filled with ordinary people who applied what they knew to a very finely balanced question and delivered a result which may or may not be right.

Just because you don’t want to risk gambling a huge sum of money on their guess doesn’t make democracy fundamentally broken, it just means recognising that people didn’t become experts on heavenly bodies just because they were asked a question on them.

In the same way that people didn’t become experts on economics, immigration, law, trade, etc. just because they were asked to vote on the UK’s future in the EU.  The strength of their opinions isn’t proportionate to the depth of their knowledge…and I’m talking about people on both sides of the debate here.  A tiny number of people on each side of the campaign had a realistic view of what their answer entailed and the rest of us guessed, and we now find far, far more than £1,000,000 riding on…well, riding on whether, as Chris would ask: