Consider the Monty Hall problem. Let's label the door with the car behind it a and the other two doors b and c. In the game the contestant chooses a door and then Monty chooses a door, so we can label each outcome as 'contestant followed by Monty ', e.g. ab means the contestant chose a and Monty chose b.

Question

(a) Suppose the contestant's strategy is to switch. List all the outcomes in the event 'the contestant wins a car '. What is the probability the contestant wins?
(b) Redo part (a) with the strategy of not switching.

Answer 1

The Monty Hall problem involves a game show scenario with three doors: behind one door (let's say door A) there is a car, and behind the other two doors (B and C) there are goats. The contestant picks one door, then the host (Monty), who knows where the car is, opens one of the other two doors (always a door with a goat). The contestant then has the option to either switch to the remaining unopened door or stick with their initial choice.

Let's consider both strategies: switching and not switching.

(a) Contestant's strategy is to switch.

Outcomes when the contestant wins a car:

1. Contestant chooses A (car behind A):

   • Monty opens B (goat behind B).

   • Contestant switches to C (goat behind C): (Not winning)

   • Monty opens C (goat behind C).

   • Contestant switches to B (goat behind B): (Not winning)

2. Contestant chooses B (goat behind B):

   • Monty opens C (goat behind C).

   • Contestant switches to A (car behind A): (Winning)

   • Monty opens A (not possible since A has the car).

3. Contestant chooses C (goat behind C):

   • Monty opens B (goat behind B).

   • Contestant switches to A (car behind A): (Winning)

   • Monty opens A (not possible since A has the car).

From these scenarios, we can summarize the outcomes where the contestant wins a car:

• Wins when initially choosing B: Switch to A (Winning).
• Wins when initially choosing C: Switch to A (Winning).

Total Winning Outcomes when Switching:

• If the contestant initially picks B or C, they will win by switching.
• The only time they do not win is when they initially pick A (2/3 outcomes are winners).

Probability of Winning when Switching:

• Wins by switching: 2 outcomes (B & C initially) won out of 3 possible initial choices.
• Probability of winning by switching = \( \frac{2}{3} \).

(b) Contestant's strategy is to not switch.

Outcomes when the contestant wins a car:

1. Contestant chooses A (car behind A):

   • Monty opens either B or C (both doors have goats).
   • Contestant keeps A and wins: (Winning)

2. Contestant chooses B (goat behind B):

   • Monty must open C (goat behind C).
   • Contestant keeps B and loses: (Not winning)

3. Contestant chooses C (goat behind C):

   • Monty must open B (goat behind B).
   • Contestant keeps C and loses: (Not winning)

Total Winning Outcomes when Not Switching:

• The only winning situation occurs when the contestant initially picks A.

Probability of Winning when Not Switching:

• Wins by not switching: 1 outcome (only when initially choosing A) out of 3 possible initial choices.
• Probability of winning by not switching = \( \frac{1}{3} \).

Summary:

• Probability of winning when switching = \( \frac{2}{3} \).
• Probability of winning when not switching = \( \frac{1}{3} \).