Question

4. An Uber driver only provides service in city A and city B dropping off passengers and immediately picking up a new one at the same spot. He finds the following Markov dependence. For each trip, if the driver is in city A, the probability that he has to drive passengers to city B is 0.25. If he is in city B, the probability that he has to drive passengers to city A is 0.45

(a) What is the 1-step transition matrix? (Let 1 = City A and 2 = City B) (5 points)

(b) Suppose he is in city B, what is the probability he will be in city A after two trips? (5 points)

(c) After many trips between the two cities, what is the probability he will be in city B? (6 points)]

Answer 1

4.

a.

1 step transition matrix :

	1 (drop at A)	2 (drop at B)
1 (pick at A)	1-0.25 = 0.75	0.25
1 (pick at B)	0.45	1-0.45 = 0.55

b.

P(city A after 2 trips | in city B currently) = P(drop at A | pick at B)*P(drop at A | pick at A) + P(drop at B | pick at B)*P(drop at A | pick at B)

= 0.45*0.75 + 0.55*0.45

= 0.585

P(city A after 2 trips | in city B currently) = 0.585

c.

long term :

p(A) + p(B) = 1

p(A) = 1 - P(B)

P(A) = 0.75*P(A) + 0.25*P(B) {from transition matrix first row}

put P(A) = 1 - P(B) in P(A) = 0.75*P(A) + 0.25*P(B)

1 - P(B) = 0.75*(1 - P(B)) + 0.25*P(B)

1 - P(B) = 0.75 - 0.75*P(B) + 0.25*P(B)

0.5*P(B) = 0.25

P(B) = 0.5

After many trips probability he will be in city B = 0.5

(please UPVOTE)