Monty Hall Problem

The Puzzle: You are on a game show. There are 3 doors:

• Behind one door is a Car.
• Behind the other two are Goats. You pick a door (say, Door #1). The host (Monty Hall), who knows what’s behind the doors, opens another door (say, Door #3), which has a goat. He then asks: “Do you want to switch to Door #2?” Is it to your advantage to switch?

1. The Paradox

Most people intuitively think since there are 2 doors left, the probability is 50-50. They are wrong. You should ALWAYS switch.

2. The Math

1. Initial State: Your door has a 1/3 chance of winning. The other two doors combined have a 2/3 chance.
2. The Reveal: Monty opens a goat door from the group you didn’t pick.
3. The Result: The 2/3 probability that was shared by Doors #2 and #3 now “collapses” entirely onto Door #2.

• Stay: 33.3% chance.
• Switch: 66.7% chance.

3. Visualization for 100 Doors

Imagine 100 doors. You pick one (1% chance). Monty opens 98 doors that all have goats. Only your door and one other door remain.

• Do you still think it’s 50-50? Of course not. The other door almost certainly contains the car.

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Interview-Focused Questions

Key Takeaway

Stay or Switch? Always Switch. You double your chances from 1/3 to 2/3.