1

Problem

In the Monty Hall problem, a contestant is asked to choose one of three doors. Behind one door is a car, and behind the other two are goats. After the contestant makes their choice, the host, who knows what’s behind each door, opens one of the remaining doors to reveal a goat. The contestant is then given the option to stick with their original choice or switch to the other unopened door. What is the probability of winning the car if the contestant switches doors?

2

Step

Initial Setup:

→ The contestant selects one door out of three

→ One door has a car behind it, and the other two doors have goats behind them

3

Step

Contestant’s Initial Choice:

→ Let’s say the contestant chooses Door 1

4

Step

Host’s Action:

→ The host, who knows what’s behind each door, then opens one of the remaining doors that has a goat behind it

→ Let’s assume the host opens Door 2 to reveal a goat

5

Step

Contestant’s Decision Point:

→ Now, the contestant is faced with a decision: stick with the original choice (Door 1) or switch to the other unopened door (Door 3)

6

Step

Probability Analysis:

The contestant sticks with the original choice (Door A) If the car is behind Door A (chosen by the contestant initially), then the contestant wins by sticking with Door A Probability of winning = \$1/3\$

7

Step

The contestant switches to the other unopened door (Door C):

→ If the car is behind Door B (not chosen by the contestant and revealed by the host), then switching to Door C (the remaining unopened door) results in winning the car

→ If the car is behind Door C, switching to Door C results in winning the car

8

Step

Probability Calculation:

Initially, the probability of choosing the car correctly with one choice out of three doors (Door A) is \$1/3\$

9

Step

After the host reveals a goat behind Door B, the remaining probability mass (probability distribution) shifts to Door C because the car must be behind either Door A or Door C.

10

Solution

Therefore, the probability that the car is behind Door C (and hence the contestant wins by switching to Door C) is 1− \$1/3\$ = \$2/3\$.

11

Sumup

Please summarize steps

Choices

12

Choice-A

This option is incorrect because \$1/3\$ is not the correct probability given the updated information after the host reveals a goat

Wrong \$1/3\$

13

Choice-B

This option is correct because \$2/3\$ is the correct probability of winning the car if the contestant switches doors in the Monty Hall problem

Correct \$2/3\$

14

Choice-C

This option is incorrect because \$3/1\$ is not a valid probability and does not accurately reflect the probabilities associated with the Monty Hall problems

Wrong \$3/1\$

15

Choice-D

This option is incorrect because the probability remains the same at \$3/2\$ even after the host reveals a goat. Sticking with your original choice keeps you at that \$3/2\$ chance

Wrong \$3/2\$

16

Answer

Option

B

17

Sumup

Please summarize choices