Yesterday Roy Meadow admitted that his errant statistics “misled and confused a lot of people”. Well, yes, in the sense that saying Michael Portillo has frequent sexual intercourse with stray animals would mislead people into thinking that Michael Portillo has frequent sexual intercourse with stray animals — i.e. it’s completely untrue. And today Ben Goldacre gives another brief statistics lecture, this time about funnels, which are new to me.

All of which allows me to write about another important thing about statistics which you really need to know, and which should be seen as a precursor to my earlier piece on multiplying probabilities of independent events.

The lesson is this: if two events are random and independent, then the probability of one won’t be affected by the outcome of the other. And if you think the probability of something is changing, then you’re also saying the events are not independent.

Let’s take an example. Let’s say you’re tossing an ordinary coin. Now you and I know perfectly well that the chance of getting heads is 1 in 2. We’ll write this as 1/2 because (a) it’s more helpful and (b) it’s correct. Half the time you’ll get heads.

Now, let’s say you’ve just tossed this coin nine times in a row and it’s come up heads every time. That’s pretty impressive — and remember, this is just an ordinary coin, no tricks. So here’s a question for you: what are the odds of getting another heads on your next flip?

Go on, close your eyes and have a guess before you read on. Are the odds good? Bad? Is it likely? Unlikely?

Okay, I’ll tell you: it’s 1/2 once again. Evens.

Now you might be surprised, espcially since getting ten heads in a row is really, really unlikely. And this is the root of the confusion.

We asked what the chance was of getting one heads. We didn’t ask about the chance of getting ten heads in a row. And that’s not a trick — there was a very good reason we didn’t ask about getting ten heads in a row. It’s because we don’t need ten heads. We’ve already got nine of them; we only need one more.

Look at it another way. The chance of getting a head on one coin toss is 1/2. If you’re gearing up to toss the coin a tenth time and you’re saying the odds are much worse than 1/2, then you’re saying something has changed about the coin, or the tossing. You’re effectively saying that something about the previous nine tosses have had an influence over the forthcoming toss. You’re saying successive tosses are not independent but dependent. Maybe you’re saying your hand would be unsteady and influence a negative outcome — but you know you can’t influence the toss of a coin at the best of times. Maybe you’re proposing a very strange invisible force that is special to coins. Clearly these are silly ideas. No previous flip of the coin is going to influence any future one. Successive coin tosses are independent. The odds remain at 1/2.

Naturally, this is true for other independent, random events. Rolling a dice, winning the lotto. Each of these is an independent event.

Let’s take the example of the lotto — not winning it, but a certain number cropping up. The nice people at Camelot keeps track of which numbers crop up when. Suppose the 33 ball gets picked three weeks in a row, but 22 hasn’t come up for six months. Should you pick the 33 ball or the 22 ball?

The answer is: it doesn’t matter. They have the same odds of cropping up again. That’s because the odds of picking one ball one week is not affected by whether it, or another ball, was picked any time previously.

Did you think there was a difference? Because if so, you’re talking about either a conspiracy among the Camelot employees, or a really mysterious force that affects lotto balls and was previously unknown to physicists. Hey, you could be the next Einstein! Or you could be mistaken.

So there we go. I’ll repeat the lesson. If two events are random and independent, then the probability of one won’t be affected by the outcome of the other. And if you think the probability of something is changing, then you’re also saying the events are not independent.

Thank you for your time. Copies of the lecture notes are available at the back. I’m off to look into Ben Goldacre’s funnels.

Tags: maths, probability, statistics