3. Hoani likes to jog around his maunga (mountain). A simplified map showing the

Question

3. Hoani likes to jog around his maunga (mountain). A simplified map showing the nine tracks (e.g.the track that goes from A to B) and seven decision points (A, B, C, D, T, V, W) is shown below. He always starts at decision point A and finishes either at decision point C or back at decision point A. Although they are different lengths, the varying terrain means that each track takes him 15 minutes to run Some days Hoani uses fair dice to determine his run. According to the following rules Every track can be traversed in both directions (eg A to B and B to A) . At every decision point he is equally likely to choose any track including the one just ran along When he reaches decision point C or returns to A he stops running . After one hour if he is still running he will stop choosing routes at random and take the quickest route back to decision point A Let A, be the event that Hoani is at decision point A after running for 15 x i minutes Define events Bi, Ci, D,, T, V and Wi i a similar manner Note: To gain full marks you must express the required probabilities in terms of these events as well as giving the correct numerical answer (a) What is the probability that Hoani begins by running to decision point D? (b) If Hoani runs first to decision point D, what is the probability that he then runs to decision point C? c) What is the probability that after running for 30 minutes Hoani is at decision point C? d) What is the probability that after running for 30 minutes Hoani is at point decision V? (e) What is the probability that Hoani only runs for 30 minutes? (f) What is the probability that after running for 30 minutes Hoani is at decision point D? (g) If Hoani is at decision point D after running for 30 minutes, what is the probability that he exits after 45 minutes? 19 marks

Explanation / Answer

(a)
Probability that Hoani begins by running to decision point D = Probability to choose the path AD
= P(D1) = 1/3 (As, there are 3 routes possible from A and each route has equal probabilities)

(b)
Probability that Hoani runs to decision point D from point C = Probability to choose the path DC
= P(C2 | D1) = 1/4 (As, there are 4 routes possible from D and each route has equal probabilities)

(c)
Probability that after running 30 minutes, Hoani is at decision point C = P(C2)
= P(C2, D1) {As, C can be reached only via D}
= P(C2 | D1) * P(D1)
= (1/4) * (1/3) = 1/12

(d)
Probability that after running 30 minutes, Hoani is at decision point V = P(V2)
= P(V2, B1) + P(V2, T1) {As, V can be reached only via B or T from A in 2 steps}
= P(V2 | B1) * P(B1) + P(V2 | T1) * P(T1)
= (1/2) * (1/3) + (1/3) * (1/3) = (1/6) + (1/9)
= 5/18

(e)
Probability that Hoani runs only for running 30 minutes
= Probability that after running 30 minutes, Hoani is at decision point C or A
= P(C2) + P(A2)

From (c), P(C2) = 1/12

P(A2) = P(A2, D1) + P(A2, T1) + P(A2, B1)
= P(A2 | D1) * P(D1) + P(A2 | T1) * P(T1) + P(A2 | B1) * P(B1)
= (1/4) * (1/3) + (1/3) * (1/3) + (1/2) * (1/3)
= (1/12) + (1/9) + (1/6)
= 13/36

Probability that Hoani runs only for running 30 minutes
= P(C2) + P(A2)
= (1/12) + (13/36)
= 4/9

(f)
Probability that after running 30 minutes, Hoani is at decision point D = P(D2)
= P(D2, T1) {As, D can be reached only via T from A in 2 steps}
= P(D2 | T1) * P(T1)
= (1/3) * (1/3) = 1/9

(g)
If Hoani is at decision point D after running for 30 min, probability that he exits after 45 min
= Probability that he reaches C or A in next step
= P(A3 | D2) + P(C3 | D2)
= (1/4) + (1/4)
= 1/2

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