The Monty Hall Problem: Should You Switch?
The Monty Hall problem reveals how conditional information changes probability: after the host always opens a losing door, switching has a 2/3 chance of winning while staying has a 1/3 chance.
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When one losing door is opened after your first choice, should you stay with your original door or switch?
A game-show choice seems simple, but one small decision can turn an apparent 50–50 guess into a much better strategy.
After the host opens a losing door, intuition says the two remaining doors must be equally likely—but is that intuition correct?
Show repeated simulated rounds, track wins for staying versus switching, and let learners compare the resulting frequencies with the initial 50–50 prediction.
Switching wins about two out of three times, while staying wins about one out of three, because the host’s action gives new information about the door you initially chose.
Once two doors remain and the car is behind one of them, the odds seem 50–50, so switching should not improve the chance of winning.
- The historical origins and names associated with the problem
- More complex variants involving randomized host behavior or multiple prizes
- A formal derivation using Bayes’ theorem
- 01Three Doors, One Car, and a ChoiceslideQuestion
Introduce the classic setup: a car is hidden behind one of three doors, goats behind the other two, and the learner chooses a door without opening it.
- The car is equally likely to be behind any door before the choice
- The learner selects one door
- The host will later open a different door
- 02Make Your First CommitmentinteractivePrediction
Let the learner independently choose whether they would stay or switch after seeing a losing door opened, then reveal the chosen strategy before any explanation.
- Commit to one strategy
- Treat the two remaining doors as equally likely at first glance
- The choice is a prediction, not a final answer
- 03What Does the Host Know?slideEvidence
Show a clear diagram of the three possible locations of the car and the host’s behavior: he always opens a losing door and never opens the player’s selected door.
- The host cannot reveal the car
- The host’s choice depends on the initial selection
- The host’s action is not random from the learner’s perspective
- 04Run the Game Many TimesinteractiveEvidence
Simulate many rounds and display separate results for staying and switching, including counts and percentages that update as the number of rounds increases.
- Run enough rounds for stable patterns
- Compare switching and staying frequencies
- Notice that the results settle near 2/3 and 1/3
- 05Why the Odds Are Not 50–50slideExplanation
Explain the three equally likely starting cases. In two cases the car is behind a door the host can safely leave closed, while in one case it is behind the learner’s original door. Switching wins in two of the three cases.
- The initial door wins in one of three equally likely cases
- The host’s action does not give the original door extra probability
- The unopened alternative receives the other two cases
- 06Test the Strategy With 100 DoorsinteractiveTransfer
Transfer the idea to a changed game: one prize is behind one of 100 doors, the host opens 98 non-prize doors, and the learner can stay or switch. Ask the learner to predict the better strategy before revealing the result.
- The same information principle applies
- Switching becomes dramatically better as the number of doors increases
- The host’s revealed information is the key to the advantage
- 07When Would the Answer Change?slideBoundary
Show that switching is not automatically best if the host can open a door at random, reveal the car, or behave independently of the player’s choice. Distinguish the classic rules from altered versions.
- The conclusion depends on the host’s behavior
- A host who may reveal the car changes the information structure
- The classic puzzle assumes the host knows where the car is and always avoids it
- 08The Best Move Is to SwitchslideResolution
Return to the driving question and state the resolved conclusion directly: with the standard rules, switching wins about 2/3 of the time and staying wins about 1/3.
- Stay: 1/3 probability of winning
- Switch: 2/3 probability of winning
- The host’s deliberate choice supplies the information that creates the advantage
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