Popular Science Feature - Ferraris and Goats

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Feature - Ferraris & Goats - Brian Clegg

Probability is one of the most useful and everyday applicable aspects of maths, yet the human mind is particularly bad at assessing probabilities. In this short article we look at a little game with a surprising outcome.

Before getting into the feature proper, I ought to point out that, when I wrote this piece I hadn't read Mark Haddon's The Incident of the Dog in the Night, in which the problem appears with the same phrasing. (The underlying "game", as mentioned below, dates back at least 100 years.) I first came across the Ferraris and goats version after the Times published it. This was either in the late 80s, predating the origin that Haddon mentions, or immediately after the US publication Haddon refers to (memory is a little weak going back that far!)

In the delightful children's book on probability Do You Feel Lucky (recommended even for adults who haven't got a grip on the subject, and that's most of us!), the author Kjartan Poskitt outlines a tricky little game (see page 162) involving three cards, one with spots on both sides, one blank on both sides and one that has both markings. It featured, though Poskitt didn't mention it, as a biased game on Mississippi riverboats, but it isn't the best way of representing the problem to baffle the player.

In this feature I want to describe a different presentation of exactly the same probability game that a few years ago baffled readers of the London Times so much that indignant letters were written by professors, denying the truth of the outcome. When I'm not writing popular science books I provide training in creativity for business people. I've tried this problem out on business brains, many of them with higher degrees, for many years - and the majority of them have always been fooled by it.

Imagine you were taking part in a game show. You've won through to the last round, which is a game of chance. There are three doors - all you have to do is to choose which door you want to open - Door 1, Door 2 or Door 3. Behind one door is a Ferrari. Behind each of the other two is a goat. Let's assume that you are normal enough to want a Ferrari.

After a moment's indecision you plump for a door - let's say it's Door 2. The gameshow host nods, knowingly. "Okay," he says, "I'm going to give you a final chance to change your mind. And I'm even going to help you." He opens Door 3 and shows you there's a goat behind it. "Now," says the host, "do you want to stick with the door you chose, or do you want to change to another door."

The question I ask you is subtly different. Should you stick with the door you chose, should you change (presumably to Door 1), or doesn't it matter in probability terms whether you stick or change?

Think about it a little while before reading on.

Don't cheat, think about it.

Most people reckon it doesn't matter, and their logic - very sensible it is too - goes something like this. The gameshow host eliminated one door. So there are two doors left, one with a goat behind it, one with a Ferrari. There's a 50:50 chance that you've got the right door. So it doesn't matter if you change or not.

If that's what you thought, you are in the majority. And you are wrong - because you can't separate the final decision from the whole game. Go back to the start. What was the chance you had the right door? 1 in 3. In two cases out of three you would have picked a goat. That holds true whatever. Then the host helps you out by showing you that one of the other doors definitely has a goat behind it. So, in fact you should always change your choice to the other door that wasn't opened. That way there's a two in three chance that you win.

You may find that explanation unsatisfactory. Plenty of people do. If so, here is the spotty card version, which is easier to believe but exactly the same problem.

There are three cards. One with a spot on either side, one blank on either side, one with spot on one side and blank on the other. A card is selected at random and put on the table. It has a spot on the face you can see. Your opponent says "I'll make a bet with you. Obviously this isn't the card with no spots. It is either the card with spots on both sides, or a spot on one side. So there's a fifty-fifty chance there's a spot on the other side. You pay me a dollar if there is a spot on the other side. I pay you a dollar if there isn't a spot." The person running the game will soon be in profit. Because, again there's a two in three chance they're right. The trick is that what they're actually betting on is whether the other side of the card is the same as the one you see. (If it had been blank face up, they would have bet that the other side was blank.) That's true two times out of three. So on the whole they win. With a little thought, you can see that this is exactly the same problem as the Ferraris and Goats.

Still not convinced? Drop us an email at info@popularscience.co.uk - and perhaps you'd like to undertake a little bet...

Brian Clegg is a popular science writer whose books include A Brief History of Infinity and Light Years.

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