A particle performs a random walk on a bow tie ABCDE drawn beneath on the left, where C is the knot. From any vertex its next step is equally likely to be to any neighbouring vertex. Initially it is at A. Find the expected value of:.
a) the time of first return to A.
b) the number of visits to D before returning to A
c) the number of visits to C before returning to A
d) the time of first return to A, given no prior visit by the particle to E.
e) the number of visits to D before returning to A, given no prior visit by the particle to E.