Readers as old as I am may remember one of TV’s more popular game shows, Let’s Make a Deal, and its host Monty Hall. Those keen on probability and statistics may also know of a now-classic, once in dispute, problem known as the Monty Hall problem.
The final game on each show involved a single contestant given a choice — door #1, door #2, or door #3. One of them conceals the “grand” prize — often a car, which seems to be a popular choice among producers and contestants alike. The other two doors conceal lesser prizes, sometimes even a “booby prize.” The relevant fact is that the contestant wants the grand prize, not either of the others.
After the contestant chose a door, Monty then revealed what was behind one of the other doors. He didn’t reveal the contestant’s choice, nor the grand prize, he showed one of the lesser prizes. Specifically, he showed a lesser prize which the contestant did not choose. In order to do so, Monty had to know which door concealed the grand prize — otherwise, he might accidentally reveal the grand prize door before the game was over, which would be terribly anti-climactic.
Then came a crucial choice: Monty offered the contestant the opportunity to change selection. If, for instance, you had originally chosen door #1, then Monty revealed that door #2 concealed a goat (not the grand prize), he would then ask, “Do you want to keep what’s behind door #1, or pick what’s behind door #3?”
The Monty Hall problem is: should you stay with your original selection, or switch?
To most people, it seems that switching doesn’t gain any advantage. After Monty reveals one of the booby prizes, there are two doors left, so the odds the grand prize is behind your original choice are 50-50. Hence it doesn’t matter whether you switch or not.
But that’s wrong. Originally — with no information at all — the chance you picked the right door is 1/3. What Monty does afterward doesn’t change that. The chance the grand prize is behind one of the other doors is 2/3.
If you picked the right door, Monty can show you either of the others. But if not, he has to show you the other one which doesn’t conceal the grand prize. The fact that Monty showed you which of those it is, doesn’t alter the fact that there’s a 2/3 chance. So: switch. It gives you a 2/3 chance of winning, whereas you only have a 1/3 chance with your original choice.
I could give formulae, present logical constructions, even run computer simulations, and they would show the same thing: switch.
The result seems counterintuitive. So much so, that many had a hard time believing it, including many mathematicians. But the answer, counter-intuitive though it may be, is correct. Switch.
Yet there are those who still don’t accept the solution. Perhaps it won’t surprise you that some of them are global warming deniers. So I’ll try another approach to persuade the unconvinced.
Let’s modify Let’s Make a Deal so that the final game has not just 3 doors to choose from, but 100 of them. Only one conceals the grand prize, the other 99 hide booby prizes. You pick one.
Then, Monty shows you what’s behind 98 of the doors you didn’t choose. He knows ahead of time which door conceals the grand prize, so he never uncovers the grand prize by accident. After all is said and done, there are two doors unrevealed: your original choice and one which you didn’t choose. The other 98 doors have already been shown to hide a goat.
Now which would you choose?
If you like what you see, feel free to donate at Peaseblossom’s Closet.