A particle starts at (0,0) and moves at each step one unit in either the positive x or y direction. The probability that it will move in the x direction is p. If the probability that the particle passes through the point (3,1) is 16 times as great as the probability that it passes through the point (1,3), find p.
+23246 I am going to assume that the particle can move only either to the East (E) or to the North (N).
Going from (0,0) to (3,1) requires, in some order E x E x E x N.
Going from (0,0 to (1,3) requires, in some order E x N x N x N.
So: E x E x E x N = 16 x ( E x N x N x N)
---> E3 x N = 16 x E x N3
Dividing both sides by E x N:
---> E2 = 16 N2
---> E2 / N2 = 16
---> E / N = 4
Since the probability of E is p, the probability of N is 1 - p:
---> E / N = p / (1 - p) = 4 ---> p = 4(1 - p) ---> p = 4 - 4p ---> 5p = 4 ---> p = 4/5
