WS MoreOrLess: Risk Savvy
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What is the main topic discussed in this episode?
This is the short edition of More or Less, first broadcast on the BBC World Service.
Hello and welcome to More or Less on the BBC World Service. We're your weekly look at numbers in the news and in life, and I'm Tim Harford. I hope you're sitting comfortably and paying close attention. We are about to describe a classic brain teaser that never fails to stir controversy. Listen carefully.
What is the Monty Hall puzzle and how is it presented on the show?
Hello and welcome to Let's Make a Deal, a game show that aired on NBC in the US in the 1960s. The host, Monty Hall, yes, that's his name, is showing the contestant three doors. Behind one of the doors is the top prize, a brand new Cadillac. Behind the other two doors are goats. not what you want. The contestant has to choose between door one, door two or door three. She thinks for a while and then chooses door one. Now to make it even more tantalising, Monty Hall, the host, walks up to door three and throws it open. It's a goat. Which means the Cadillac must be behind door one or door two. Monty now turns to the contestant and offers her the chance to switch doors. But what would you do? Remember, door one and door two are still unopened.
Door three has a goat behind it. The contestant chose door one to start with, but now the host, Monty Hall, is offering her the chance of swapping. Should she do it? This puzzle features in a new book called Risk Savvy by Gerd Gigerenza, director of the Max Planck Institute for Human Development in Berlin. And I caught up with him to find out whether the contestant should swap or stick with her choice of door one and what that choice tells us about risk and uncertainty. Now, this is a fantastic and well-known among probability nerds brain teaser. So give us the standard answer, which I think people will be surprised by.
Why does switching doors increase your chances in the Monty Hall problem?
Yeah.
Yeah, most people intuitively say, no, I stick with my door. Possibly because they think it's a 50-50 chance. There are two doors, one or two. And if I switch, I regret it. So I'd rather do nothing, stick with it. The surprising answer is that in this textbook problem, the best choice is to switch.
So why would that be? I know it's counterintuitive, but the chance that the Cadillac is behind door one was originally one in three. And when Monty Hall opens door three and shows the Cadillac isn't behind door three, that chance doesn't change. There's still a one in three chance that the Cadillac is behind door one. And so the contestant can give herself a two in three chance to win by switching to door two.
How does the 100-door analogy clarify the Monty Hall probabilities?
Or let me show the logic in another way. Imagine there were 100 doors and still only one Cadillac. The contestant picks a door. Monty Hall opens 98 other doors and none of them have the Cadillac behind. And so the contestant can stick with her original door, very unlikely to be right, or she can switch to the only other door that's still closed. And if the original door didn't have the Cadillac behind it, then the other door definitely will. Well, so much for the most controversial and counterintuitive probability puzzle around. But Gerd Gigerenzer says, not so fast. Are we sure it really is a probability puzzle?
That's the typical story. But now the real twist comes. The Monty Hall problem is about risk. Everything is known.
Why does Gerd Gigerenzer argue the Monty Hall problem may not be a pure probability puzzle?
And Monty has to follow rules.
For example, the rule that he always has to offer the contestant the chance to swap. And maybe in real life, he doesn't. Maybe he's actively trying to fool the contestant. If the rules are known, then we can calculate probabilities. But if they aren't known, we can't simply solve the problem using probability theory. And it turns out that in the real game show, Monty Hall didn't always offer people the chance to swap. Sometimes he played tricks, offered them money to swap, mixed things up in all kinds of ways. This changes Let's Make a Deal from a world of risk, in which probabilities can be calculated, to a world of uncertainty, in which they can't.
In an uncertain world, it's an illusion to think that probability theory will give you the optimal answer. What we need is both probability theory for situations where we can calculate the risk, but also smart heuristics or rules of thumb and good intuition in a world where we can calculate that.
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Chapters
8 chapters
1
What is the main topic discussed in this episode?
0:00–0:43
2
What is the Monty Hall puzzle and how is it presented on the show?
0:43–2:30
3
Why does switching doors increase your chances in the Monty Hall problem?
2:30–3:16
4
How does the 100-door analogy clarify the Monty Hall probabilities?
3:16–4:08
5
Why does Gerd Gigerenzer argue the Monty Hall problem may not be a pure probability puzzle?
4:08–5:15
6
When does a situation move from calculable risk to true uncertainty?
5:15–7:22
7
How did Gigerenzer use investment examples to show the limits of optimization?
7:22–8:49
8
How likely are you to celebrate a round-number birthday on a weekend and what did listeners find?
8:49–9:31