INKWIRE(7) reasoning puzzles INKWIRE(7)
monty-hall
Three Doors and a Host Who Knows
The prize does not move, but the information does.
NAME
Problem
Three doors. Behind one is a prize; behind the other two is nothing. You pick door 1. The host, who knows where the prize is, opens door 3 to show it is empty, and offers you a choice: keep door 1, or switch to door 2.
He was always going to open an empty door, and he was never going to open the one you picked. Should you switch, does it make no difference, or does it depend on how the host decides which empty door to open when both are available?
The usual answer — it makes no difference, two doors means fifty-fifty — is wrong. The precise answer separates the case where the host's rule is known from the case where it is not.
SIGNALS
Conditions and what each one forces
| # | Condition | What it forces |
|---|---|---|
| 1 | Prize placed uniformly at random before you choose | Door 1 holds the prize with probability 1/3, and the other two together hold it with probability 2/3 |
| 2 | The host knows where the prize is | His action is not random; it is a deterministic function of the prize location and your choice, and therefore carries information about the prize |
| 3 | The host always opens an empty door | If the prize is behind door 1 he can open either 2 or 3; if it is behind door 2 he must open 3; if behind door 3 he must open 2 |
| 4 | The host never opens your chosen door | Your original pick survives untouched, so its probability cannot be updated upward; the 2/3 stays on the two doors you did not choose |
| 5 | He opened door 3 and it was empty | Switching to door 2 wins in exactly those worlds where the prize was behind door 2, which is 1/3 of all worlds, against 1/3 × 1/2 = 1/6 for staying |
EXAMPLES
Method
Write out the host's decision rule, enumerate the worlds consistent with what you observed, and divide. The opened door is not evidence about itself; it is evidence about the rule that chose it.
[+] Worked solution 4
Count worlds. There are three equally likely placements. If the prize is behind door 1, the host may open 2 or 3; suppose he flips a coin, so this case splits into two half-worlds of probability 1/6 each. If the prize is behind door 2, he must open 3 — one world of 1/3. If behind door 3, he must open 2 — one world of 1/3.
You observed door 3 opening. That observation eliminates the world where the prize is behind door 3. The surviving worlds are: prize behind door 1 with door 3 opened (1/6), and prize behind door 2 with door 3 opened (1/3). Total probability of the observation is 1/6 + 1/3 = 1/2.
Conditioning gives P(prize behind door 1 | door 3 opened) = (1/6) / (1/2) = 1/3, and P(prize behind door 2 | door 3 opened) = (1/3) / (1/2) = 2/3. Switching wins twice as often as staying.
The rule matters. If the host always opens the lowest-numbered empty door rather than choosing randomly, then when the prize is behind door 1 he always opens door 2, so seeing door 3 open would mean the prize is behind door 2 with certainty. Same doors, same host, different rule, different answer.
Kindred entries
- [F] base-rate A vivid new detail outweighs the much larger number it should be measured against.
- [P] census Everyone already knows, and yet nothing happens until it is said aloud.
- index — /t/puzzle/
[+] Notes and objections 3
- Why does the host's knowledge change anything?
- Because it makes his choice non-random. A random reveal would be uninformative about the prize; a reveal constrained by knowledge is a message, and messages update probabilities.
- What if the host opens a door only when you look unsure?
- Then the fact that he opened one at all is additional evidence, and the 2/3 figure no longer applies without specifying the full rule. This is why the puzzle is a lesson about conditioning on rules, not about doors.
- Does the answer survive if you picked door 2 instead?
- Yes, with the labels permuted. Whichever door you pick keeps probability 1/3, the two you did not pick keep 2/3 between them, and the host's reveal concentrates all of that 2/3 onto the single remaining unopened door.
Scope — Puzzle presented for practice; the conditional probabilities above are exact under the stated host rule.