Real life · Conditional probability
The Monty Hall Problem Explained: Why You Should Switch Buckets
The short version: if the host knows where the prize is and always opens an empty option you didn't pick, switching wins 2 times in 3 and staying wins 1 in 3. Your first pick was right only a third of the time, and the host's choice can't change that. With 100 buckets and 98 opened, switching wins 99%. But if the host opens one at random and it happens to be empty, it really is 50/50. The answer depends on the rule behind what you were shown, not on what you see.
Cousin Kevin runs the "Find the Voucher" stall at the school fete. Under one of three upturned buckets is a $50 voucher for the butcher's. You pick a bucket. Kevin, who knows where the voucher is, lifts one of the other two, always an empty one, and asks whether you'd like to switch.
Uncle Gary picked bucket one. Kevin lifted bucket three: empty. "Switch?" Gary refused. "I never switch. It's a matter of loyalty. Also, it's fifty-fifty now. Two buckets, one voucher." Bucket one was empty. The voucher was under bucket two.
Gary has since played eleven times, stayed eleven times and won four. He describes this as "about half". The headteacher has asked whether he'd consider letting some children have a go.
Why it isn't fifty-fifty
Forget the lifted bucket for a moment and look at Gary's first pick. With three buckets, it's right 1 time in 3. Nothing Kevin does afterwards can change that, because Kevin will always lift an empty bucket, whatever Gary picked.
- One time in three, Gary picked the voucher. Kevin lifts either empty bucket. Switching loses.
- Two times in three, Gary picked an empty bucket. Kevin must lift the other empty one, because he can't lift the voucher. The bucket left over has the voucher. Switching wins.
So staying wins 1 in 3 and switching wins 2 in 3. Gary's eleven games, four wins, are almost exactly what staying should give him.
The 100-bucket version
If it still feels wrong, make it bigger. A hundred buckets, one voucher. Gary picks one. Kevin, who knows, lifts 98 empty buckets and leaves one other bucket standing. Stay or switch?
Gary's first pick had a 1 in 100 chance. The one bucket Kevin carefully avoided lifting, out of 99, has the other 99 in 100. Nobody would stay. With three buckets it's the same logic; it just doesn't shout.
It depends on what the host knows
Now change one thing. Kevin doesn't know where the voucher is. He lifts one of the other buckets at random, and it just happens to be empty. In that case it really is fifty-fifty, and switching doesn't help.
Same buckets, same empty bucket in front of you, different answer. The difference is the rule that produced what you saw. When Kevin knows, his choice carries information about where the voucher is. When he's guessing, it doesn't. That's conditional probability: what you should believe depends on how the evidence was chosen, not just on the evidence. (Lesson 11 of our course works through it with a car alarm.)
This puzzle is named after Monty Hall, host of the American game show Let's Make a Deal. In 1990, the columnist Marilyn vos Savant explained in Parade magazine that you should switch, and received thousands of letters telling her she was wrong, many of them from people with doctorates. She wasn't.
Chance of winning the voucher
Try it at home
1. Three mugs, one biscuit. One person hides the biscuit and plays Kevin: they always lift an empty mug you didn't pick.
2. Play 30 rounds staying and 30 switching. Write down every result. Expect about 10 wins staying and about 20 switching. The more rounds you play, the closer it gets.
3. Then play with a host who guesses. Throw out the rounds where they lift the biscuit by accident. Now staying and switching come out about even.
Where it costs you money
Whenever someone shows you information, ask how they chose what to show. A fund company that displays its five best funds is a host who knows where the empty buckets are: the funds on the page are real, but they were picked because they did well. The rule behind the selection tells you more than the selection itself. And "I've already chosen, so I'll stick" is loyalty, not analysis.
The moral
Aunt Linda set up three mugs and a biscuit at the kitchen table and hosted 60 rounds. Staying won 9 of 30. Switching won 21 of 30. Gary studied the results and announced that the game was rigged, because Linda knew where the biscuit was the whole time. "Yes, Gary," said Linda. "That's the whole point." Gary has gone back to the fete. He now switches every time, including when Kevin forgets to lift a bucket.
The stall and its voucher are fictional; the probabilities assume a host who knows the prize and always opens an empty option you didn't choose. Educational content, not financial advice.