Question

A man is trapped in a room at the center of a maze. The room has three exits. Exit 1 leads outside the maze after 3 minutes, on average. Exit 2 will bring him back to the same room after 5 minutes. Exit 3 will bring him back to the same room after 7 minutes. Assume that every time he makes a choice, he is equally likely to choose any exit. What is the expected time taken by him to leave the maze?

Hint:

Let X = time taken by the man to leave the maze from this room.

Let Y = exit he chooses first. So Y belongs in {1, 2, 3}.

Calculate the conditional expectation of time taken to leave the maze given that he chose each of the exits. Then use these conditional expectations to calculate the expectation of time taken to leave the maze.

Answer 1

Answer:

Given

Exit 1 E(X| x = 1) = 3

exit2 E(X|x = 2) = 5+ E(x)

exit 3 E(X|x = 3) = 7 + E(x)

conditional expectation of time taken to leave the maze E(x) = P(x) * E(X|x)

= P(x =1)* E(X|x =1) +P(x = 2) * E(X|x = 2) + P(x = 3) * E(X|x = 3)

= (1/3)*3 + (1/3) * (5 + E(x)) + (1/3)* (7 +E(x))

= 15 minutes.

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