A mouse is in the center room of a maze with three exit doors: Door A will lead

Question

A mouse is in the center room of a maze with three exit doors: Door A will lead back to the center room after 6 minutes, door B will lead back to the center room after 18 minutes, and door C will lead to the outside (and a treat) after 12 minutes.

(a) At what time do you expect the mouse to leave the maze if it chooses any of the exits equally likely every time? (i.e. a mouse with no memory or learning capability)?

(b) At what time do you expect the mouse to leave the maze if it chooses any of the exits equally likely every time? (i.e. a mouse with no memory or learning capability)?

(c) A mouse with perfect memory chooses between the three exits only the very first time. At all other other times, this mouse will not repeat any previous decision. At what time do you expect the mouse the leave?

Explanation / Answer

a) Let the expected time of mousing leaving be T

T = (1/3)(6+T)+(1/3)(18+T)+(1/3)(12)

3T = 2T+(6+18+12)

T = 36

expected time of mousing leaving is 36

b)Same question as (a)

c)

1)Mouse can take door C directly to get out

Mouse can take door A and then C to get out

Mouse can take door A , then B and then C to get out

Mouse can take door B and then C to get out

Mouse can take door B , then A and then C to get out

Expected time = (1/3)*12+(1/3)(6+(1/2)*12+(1/2)*(18+12)) +(1/3)(18+(1/2)*12+(1/2)*(6+12)) = 24

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