Batman chases the Joker around the vertices of a square. At each time step, Batm

Question

Batman chases the Joker around the vertices of a square. At each time step, Batman steps clockwise with probability p (0, 1), and anticlockwise with probability 1p, while the Joker stays where he is with probability q [0, 1], and steps clockwise with probability (1 q)s and anticlockwise with probability (1 q)(1 s), where s (0, 1). Batman catches the Joker if they reach the same vertex at the same time. All steps are taken independently of previous steps. Starting from opposite corners of the square:

(a) if q = 0, find the expected time until Batman catches the Joker, and for fixed p, find the maximum possible value for this quantity (i.e. optimise over s).

(b) When p = 1/2, find the expected time until Batman catches the Joker.

Explanation / Answer

a) When q = 0,

Expected time = p(1- p ) + (1 - q). s . (1-q)(1-s)

= p(1-p)

b) when p =1/2 ,

Expected value = 1/2 (1 - 1/2)

= 1/2 X 1/2

=1/4

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