Bob, Alice and Carol are playing frisbee. Bob always throws to Alice and Alice a
ID: 3249090 • Letter: B
Question
Bob, Alice and Carol are playing frisbee. Bob always throws to Alice and Alice always throws to Carol. Carol throws to Bob 2/3 of the time and to Alice 1/3 of the time. (a) Construct a transition diagram, clearly labelling the probabilities, to represent this situation. (b) Construct a transition matrix to represent this situation. (c) If Carol throws the frisbee initially, what is the probability that Bob will have the frisbee after the third throw? (d) In the long run, what percentage of the time do each of the players have the frisbee?Explanation / Answer
(a) and (b)
The transition probability matrix is given by :
P =
(c) P3 = P2 * P
P2 = P * P =
P3 =
The probability that Bob will have the frisbee after the third throw is 0.22.
(d)
(1) q1 = 0.67q3
(2) q2 = q1 + 0.33q3
(3) q3 = q2
and (4) q1 + q2 + q3 = 1
Solving equations (1),(2) and (4) , we get
q1=0.26 , q2=0.37 and q3=0.37 , i.e Bob 26% , Alice 37% , Carol 37% of the time have the frisbee.
Bob Alice Carol Bob 1 Alice 1 Carol 0.67 0.33