Question

Oh no! Thomas is lost in the G.G. Brown Labyrinth building after an exam and is trying to get out. There are four identical hallways leading away from him. One of the hallways will take 12 minutes to walk and will lead him out. However the other three hallways will take 10 minutes, 10 minutes, and 40 minutes respectively, to walk. At the end of those three hallways, Thomas falls through a trapdoor back to where he started. Assume the following:

• Thomas can choose only one hallway at a time and cannot turn around

• When Thomas falls through a trapdoor he cannot tell which hallway he chose last

• The hallways are indistinguishable aside from their outcome

Let T be a random variable for how long it will take Thomas to escape. What is E(T)?

Hint: Consider three events, H₁, H₂, H₃ where Thomas chooses the escape hall, one of the 10-minute halls, or the 40-minute hall, respectively. Note that H₁, H₂, and H₃ cover all possible cases for the first choice of hallway and do not overlap. The law of total expectation states that: E(T) = E(T | H₁) P(H₁) + E(T | H₂) P(H₂) + E(T | H₃) P(H₃)

Answer 1

ANSWER::

Since each of the 4 hallways is equally likely then

P(H1)=1/4

P(H2)=1/2

P(H3)=1/4

Now, if he chooses H1 he escapes after 12 minutes then

E(T|H1)=12

If he chooses H2 then he wastes 10 minutes and then he is again in the same starting

position so he expects to escape in E(T) minutes, then

E(T|H2)=10+E(T)

Analogously, if he chooses H3 then he wastes 40 minutes and then he is again in the same starting

position so he expects to escape in E(T) minutes, then

E(T|H3)=40+E(T)

Therefore

So we have

Thereofre he expects to escape in 72 minutes

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