Question

A mouse is put into a linear maze with 6 cells, with Cell 0 on the left end where the mouse will receive a shock, and Cell 5 on the right end, where the mouse will find food. The four cells in between are numbered 1 to 4. Assume that from any of the cells 1 to 4, the mouse moves right or left with probability p and q = 1 ? p

a) Starting from Cell 3, find the probability that the mouse finds food before being shocked. (b) Assuming that p = q = 0.5, and that the starting cell is Cell 2, find the probability that the mouse finds food before being shocked

Answer 1

for mouse to find food before shock

let the stat at k be g(k) so we have

g(k) =q*g (k+1) +p*g(k-1) 1<=k<=4

at g(5) we will have g(5)= 1 and g(0) =0

so

g(0)=0

g(1) =q g(2)

q(2) = q g(3) + p g(1)

g(3) =q g(4) + pg(2)

g(4) = q + p g(3)

solving

subtituting one afte tthe other we will get

g(4) =2q/(1+pq)

g(3) = (1-pq) g(4) /2p

= (1-pq)q/(1+pq)p

g(2) =q g(3) /1-pq

=1/((1+pq)q)

at .5, .5

we have p(2) =2/5