3 oeople - john joan and kim throw a ball to each other. there is a probability of 1/3rd that johnwill throw it to joan. probability of 1/2 that joan wil throw it to kim. probability of 1/4th that kim will throw it to john.

A. show in a Markov chain as a transition matric.
B. assuming the ball starts with joan what is the probability she will have it back in 2 throws.

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I am unsure out the mattrix. for he ball being thrown back to her i thought it woud be 1/5

A. To represent this scenario as a Markov chain, we can define the states as each person having the ball. Let's use the notation J for John, A for Joan, and K for Kim. The transition matrix, P, can be constructed based on the given probabilities:

P = [[0, 1/3, 2/3],
[1/2, 0, 1/2],
[1/4, 3/4, 0]]

Each row i represents the probabilities of transitioning from state i to states J, A, and K respectively.

B. To find the probability that Joan will have the ball back in 2 throws, we need to calculate the probability of transitioning from A to K and then from K back to A after those two throws. We can do this by multiplying the corresponding transition probabilities from the transition matrix P.

The probability of transitioning from A to K in one throw is given by P[A][K] = 1/2. Then, the probability of transitioning from K back to A in one throw is given by P[K][A] = 3/4.

To find the overall probability of Joan having the ball back in 2 throws, we multiply these probabilities together:

P[2] = P[A][K] * P[K][A]
= (1/2) * (3/4)
= 3/8

Therefore, the probability that Joan will have the ball back in 2 throws is 3/8.