Random Walk around six rooms,

Question

Hi, I have no idea how to do this question. From my probability I course, first year. Any hints or suggestions would be very appreciated.

Answer 1

HINT:

Let us name the rooms $R1,R2,... R6$ etc.

From $R1$,in one second it can go to $R2$ or $R4$ with $Pr = 1/2$ each
which we can write as $R1 = 1+ (1/2)(R2+R4)$

Similarly, from $R2$, in one second it can go to $R1,R3,R5$ with $Pr = 1/3$ each,
which we can write as $R2 = 1+ (1/3)(R1+R3+R5)$

You can try to write out similar equations from other rooms, I shall only write the final tricky one, viz

from $R5$ in one second it returns to $R2$ with $Pr = 1/2$ or, alas, the cat gets it

$R5 = 1 + (1/2)R2$

If you solve the system of linear equations, the value you get for $R1$ will give you the required time value