Question

2. Each time that Mary purchases a tube of toothpaste, she chooses either brand A or brand B. Suppose that the probability of switching from A to B is 2/3 and from B to A is 1/4. Compute the probability p that her third purchase will be brand B given that her first purchase was brand A. Compute also the probability q that if you see Mary shopping for toothpaste next year that she will purchase brand B eventhough you have no idea what brand she purchased first. Assume Mary’s choices of toothpaste follow a Markov Chain.

(A) p = 5/18and q = 3/11

(B) p = 150/216and q = 8/11

(C) p = 150/216and q = 3/4

(D) p = 157/216and q = 3/12

(E) p = 13/18and q = 8/11 3.

Answer 1

here transition mtrix T =
	A	B
A	1/3	2/3
B	1/4	3/4

state vector for third purchase =Xo*T² =[ 5/18 13/18]

therefore p=13/18

let fixed probability vector is W = [a 1-a]
therefore from relation: WT =W
0.333333333333333a+0.25(1-a)=a …….(1)
solving above: a=0.273

ong term probability of B =1-a =8/11

corect option is E p = 13/18and q = 8/11