Should You Switch Doors?

Noor pointed before anything opened. Slide the hidden star through every hiding place — her marker never moves.

hiding place: 1 · under my cup

place 1place 3

One marker, three hiding places.

Now look at the host. He read his card before the game began, so he already knows every hiding place.

The grey-haired fair grown-up bends his head down and reads a small coral card he holds up in one hand, while his other hand rests on the handle of one closed cupboard.

So does a hint from somebody who peeked work the same way as a hint from somebody guessing?

Different game: one key, three tins — choose a helper.

try: 1 of 6 · key stayed hidden · 0 uncovered

try 1try 6

One peeked first, one did not — watch what each can lift.

Fresh test: nine sealed rounds. Noor always points at cupboard 1.

teal marks where Noor points

the prize hides behind each cupboard exactly three times
the host always peeks first
he always opens an empty cupboard she did not point at
he always offers the change of mind

Nine sealed rounds. Noor always points at cupboard 1 first.

Which plan collects more prize stars across all nine rounds?

Here is the whole map. When her first point already sat on the prize, keeping collects. In every other round the host had to leave the prize behind the one cupboard still closed.

first point started RIGHT — 3 rounds

first point started WRONG — 6 rounds

keeping collects moving collects

But wait…

Does the better plan always win? ＋

No. In any single round the prize really can be sitting behind the cupboard Noor first pointed at, and then moving walks her straight off it. The plan is the better bet, never a promise.

What if the host had not peeked? ＋

Then the whole thing changes. A host opening a cupboard at random could reveal the prize itself. In the rounds where he happened to miss it, the two closed cupboards really would be even. The edge comes from his hint being restricted.

The same girl and grey-haired grown-up stand quietly at either end of the same three matching closed cupboards at the fair.

One blind point covers one hiding place. The other two sit under cupboards nobody chose. A host who peeked can never open the prize — so every round her first point missed, the prize had nowhere left to be but the one cupboard still closed.

psst, grown-ups →

This is the Monty Hall problem. The first pick is correct with probability 1/3, so the prize is elsewhere with probability 2/3. Because the host knows the layout and is constrained to open a losing, unchosen door, no probability mass transfers back to the original door; the entire 2/3 collapses onto the single remaining closed door. Switching therefore wins 2/3 of the time and staying 1/3. The crux is the host's constraint — a host opening uniformly at random, who sometimes reveals the prize, leaves a genuine 50/50 between the two closed doors in the surviving cases. New information only moves the odds when the information itself was filtered.