Question

Oh no! Thomas is lost in the GGBL building after an exam, and trying to get out. Thereare four identical hallways leading away from him. One of the hallways will take 12 minutesto walk and will lead him out. However the other three hallways will take 10 minutes, 10minutes, and 40 minutes respectively, to walk. At the end of those three hallways Thomasfalls 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 outcomeLetTbe a random variable for how long it will take Thomas to escape. What isE(T)?Hint:Consider three events,H1,H2,H3where Thomas chooses the escape hall, one of the10 minute halls, or the 40 minute hall, respectively. Note thatH1,H2, andH3cover allpossible cases for the first choice of hallway and do not overlap. The law of total expectationstates that:E(T) =E(T|H1)P(H1) +E(T|H2)P(H2) +E(T|H3)P(H3)

Answer 1

Answer:

Check the explanation

Step-by-step explanation:

Since each and every one of the 4 hallways is equally likely then

$P(H1)=1/4P(H2)=1/2P(H3)=1/4$

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

$E(T) = E(T|H1)P(H1) + E(T|H2)P(H2) + E(T H3)P(H3) = (12) + (E(T) +10) + (E(T) + 40) = 3+ *E(T)+5+ 10 = E(T) +18 CON A'$

So we have

$E(T) = E(T) + 18 E(T) = 18 E(T) = 4(18) = 72$

Therefore he is expected to escape in 72 minutes