The Monty Hall problem: switch, and win two times in three

A car is behind one of three doors and goats are behind the other two. You pick a door. The host, who knows where the car is, opens one of the other two and shows you a goat. Then he offers you the chance to switch to the last closed door. Most people say it makes no difference. It does: switching wins 2/3 of the time, sticking only 1/3.

“But two doors are left, so it’s 50/50”

That is what nearly everyone says, and here is the cleanest way out of it. Your first pick is a 1 in 3 guess, and nothing the host does can change what is behind it. Switching is the same as being allowed to have both of the other two doors — the host has just opened the one of them that had a goat behind it, which he can always do. Two doors beat one.

If that still feels wrong, make it bigger. With 10 doors, you pick one and the host opens 8 goats, leaving your door and one other shut. Your door was a 1 in 10 guess; the other is the door he carefully avoided opening. Switching wins 90.0% of the time.

Play it

This is the puzzle as it is usually told: the host knows where the car is and always shows you a goat.

Every game, counted

A few games prove nothing, and you should not have to play a thousand. So here are 300 games played out exactly, with every equally likely case in its true proportion. You pick a door each time; it is the car in a third of them.

The second row is the whole difference. In the 100 games where your first pick was the car, both hosts show you a goat. In the other 200, the knowing host always shows a goat too — so all 200 are wins for switching. The random host opens the car in 100 of them, and those games are thrown out; what is left is 100 games where sticking wins and 100 where switching wins. Seeing a goat by luck tells you less than seeing one by design.

Why switching works: all three cases

Say you pick door 1. The car is equally likely to be behind any of the three doors, so there are only three cases to check:

Sticking wins in one case of three and switching in the other two. The host is the whole trick. Whenever your first pick is wrong — two times in three — he has no choice about which door to open, because one of the two he could open hides the car. By opening the other, he points straight at it.

It all depends on the host

The answer comes from what the host knows and how he chooses, not from the doors. Jeffrey Rosenthal’s paper on the puzzle works through hosts who behave slightly differently, and the answers move:

• The host who opens a door at random (Rosenthal’s “Monty Fall”: he slips and knocks one open) and happens to reveal a goat. Now sticking and switching are both 1/2 — the counted games above show why.
• The host who always opens the lowest-numbered door he can (“Monty Crawl”). If he opens the higher door, he must have been forced to, so switching wins 100% of the time. If he opens the lower one, it is 1/2. Switching never does worse than sticking, which is why “always switch” is still the right rule.

One more host breaks it completely, and it needs no paper to see why: one who offers the switch only when you have already picked the car. Then switching always loses. The puzzle’s answer assumes he offers it every time, whatever you picked.

More doors, more rounds

Even if the host opens just one goat door out of many, switching still helps: with 10 doors it wins 11.25% against 10.0% for sticking. And there is a longer version with four doors and two rounds: you pick, the host opens a goat, you stick or switch; he opens another goat, you stick or switch again. MathWorld works out all four plans:

The best is stick, then switch, winning 3/4 — hold your nerve until the very end, then switch.

Where it came from

As Rosenthal recounts, the puzzle went famous when Marilyn vos Savant answered it in her Parade column of 9 September 1990. She said switch, received thousands of letters — numerous professional mathematicians among those insisting she was wrong — and was vindicated when maths classes across the country played it out with pennies and paper cups. It is older than the column: MathWorld’s references go back to a 1975 piece by Steve Selvin in The American Statistician.

Questions people ask

Should you switch in the Monty Hall problem?

Yes. Switching wins 2/3 of the time and sticking wins 1/3, provided the host always opens a door with a goat behind it and always offers the switch.

Why isn’t it 50/50 once two doors are left?

Because the host’s choice is not random: he knows where the car is and never opens it. Your first pick is right only 1/3 of the time, and every time it is wrong the car is behind the door he left shut. If the host opens a door at random and it just happens to hide a goat, it really is 50/50.

How the site works out its numbers is on the methodology page.