Question

You wake up in an empty room with three doors labeled A, B, and C. There is a paper note saying that all doors will eventually lead to the exiting hallway. Some doors will have more doors behind them. Each door has a probability ​p​ of being unlocked, independent of the state of other doors.

a) There is exactly one extra door behind door A and no doors behind doors B and C. What's the probability of reaching the exit?

b) There are exactly ​k-​ 1 doors behind door A (making a total of ​k​ doors including door A) and no doors behind doors B and C. What's the probability of reaching the exit?

2. c) The exit is reachable and there are exactly ​k​-1 doors behind door A (making a total of ​k​ doors including door A) and no doors behind doors B and C. What's the probability of being able to reach the exit through door C?

3. d) The exit is reachable and there are exactly ​k​-1 doors behind door A (making a total of ​k​ doors including door A) and no doors behind doors B and C. What's the probability of the last door behind door A being unlocked?

Answer 1

a) Since there are 3 doors and you don't know which one leads to the exit or which one will open, you are equally likely to choose any of the doors.

P(A) = P(B) = P(C) = 1/3

Probability of door being unlocked = p

P(exit) = P(A)P(A and the door behind A where both unlocked) + P(B)P(B was unlocked) + P(C)P(C was unlocked)

b) P(exit) =  P(A)P(A and all the doors behind A being unlocked) + P(B)P(B was unlocked) + P(C)P(C was unlocked)

c) P(exit through C/exit) = ?

Here,

P(exit) = (1/3)(p^k + 2p)

d) P(last door behind door A is unlocked) = p

Because as per the information given - Each door has a probability ​p​ of being unlocked, independent of the state of other doors.