Monty Hall Problem | FractionRush

🎲 The Setup

Topics: conditional probability, tree diagrams, common misconceptions

You are a contestant on a game show. There are 3 doors. Behind one door is a car. Behind the other two are goats.

Step 1: You pick a door — say, Door 1.
Step 2: The host (who always knows where the car is) opens a different door to reveal a goat — say, Door 3.
Step 3: The host offers you a choice: stick with Door 1, or switch to Door 2.

Question: Does it matter whether you switch? Is one choice better than the other?

Tree of possibilities (you always pick Door 1):

Car behind Door 1 (prob 1/3): Host opens Door 2 or 3 → Stay wins, Switch loses
Car behind Door 2 (prob 1/3): Host must open Door 3 → Stay loses, Switch wins
Car behind Door 3 (prob 1/3): Host must open Door 2 → Stay loses, Switch wins

Stay wins in 1 out of 3 cases. Switch wins in 2 out of 3 cases.

Why switching helps: When you first pick, you have a 1/3 chance of being right. The other two doors together hold a 2/3 chance. The host's reveal doesn't change your original odds — it just concentrates the 2/3 probability onto the one remaining door. Switching is like betting on both other doors at once.

📝 Questions

When you first pick Door 1, what is the probability the car is behind your door? What is the probability it is behind one of the other two doors combined? 2 marks

The host opens Door 3 to reveal a goat. The host always reveals a goat — this is guaranteed. Does this new information change the probability that the car is behind Door 1? Explain. 2 marks

If P(car behind Door 1) is unchanged after the reveal, and the car must be behind Door 1 or Door 2 (Door 3 is open), what is P(car behind Door 2) now? 2 marks

Complete the table below showing all scenarios where the car is behind each door and you always pick Door 1. Tick whether staying or switching wins each time. 3 marks

Car behind	Host opens	Stay wins?	Switch wins?
Door 1	Door 2 or 3
Door 2
Door 3

Using your table, what fraction of the time does staying win? What fraction of the time does switching win? What is the optimal strategy? 3 marks

Many people believe it "doesn't matter" because they see two remaining doors and assume 50/50. Explain clearly why this reasoning is wrong. 3 marks

Extension: Suppose there were 100 doors. You pick one. The host opens 98 doors, all revealing goats, leaving your door and one other. What is the probability of winning if you switch? What if you stay? 5 marks

Answer Key

🚪 Monty Hall

1.P(car behind Door 1) = 1/3. P(car behind Door 2 or 3) = 2/3.

2.No. The host's action was predetermined — he always reveals a goat. It gives you no new information about Door 1. Your door's probability stays at 1/3.

3.P(car behind Door 2) = 1 − 1/3 = 2/3. All the probability from Door 3 transfers to Door 2.

4.Car behind Door 1: host opens 2 or 3, Stay wins, Switch loses. | Car behind Door 2: host opens Door 3, Stay loses, Switch wins. | Car behind Door 3: host opens Door 2, Stay loses, Switch wins.

5.Staying wins 1/3 of the time. Switching wins 2/3 of the time. Always switch.

6.The two remaining doors are NOT equally likely. Door 1 was locked in at 1/3 before the host acted. The host's reveal was not random — it was constrained to always show a goat. The 50/50 assumption ignores this asymmetry.

7.P(win by switching) = 99/100. P(win by staying) = 1/100. The 1/100 chance of your original pick being right is unchanged; switching concentrates the remaining 99/100 onto the one other door.

Answer Key

Monty Hall

1.1/3 for your door; 2/3 for the other two combined.

2.No. Host always reveals a goat — no new info about Door 1. Still 1/3.

3.2/3 — all probability from opened Door 3 moves to Door 2.

4.Door 1: Stay wins. Door 2: Switch wins. Door 3: Switch wins.

5.Stay: 1/3. Switch: 2/3. Always switch.

6.The doors are not equally likely — Door 1 was fixed at 1/3 before the reveal. The host's constrained action transfers probability, not re-randomises it.

7.Switch wins 99/100; stay wins 1/100.

🚪 The Monty Hall Problem

Answer Key

🚪 Monty Hall

Answer Key

Monty Hall