A basketball player attempts a sequence of shots from the free throw line. Suppo

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A basketball player attempts a sequence of shots from the free throw line. Suppose the probability that the player makes a shot in dependent on the result of the two most recent shots. If you answer either of these questions I will give you 5 stars.

2 BASKETBALL PLAYER A basketball player attempts a sequence of shots from the free throw line. Suppose the prob- ability that the player makes a shot is dependent on the result of the two most recent shots. (1) If he made both, then the probability that he makes the next shot is 0.92; (2) If he made one, then the probability that he makes the next shot is 0.75; (3) If he missed both, then the probability that he makes the next shot is 0.4; With those information, solve the following problems: (1) Model the sequence of shots using a discrete-time Markov Chain; (2) Determine the steady state probability for the Markov Chain; (3) Given that the player missed her last shot, what is the probability that she makes her next shot? (4) [Extra Credit] Determine the player's long-term percentage of shots made; 3 STEADY STATE PROBABILITY SIMULATION In class, we have observed how the transition probability matrix of a two-state Markov chain might evolve as the number of steps increases. In this problem, you are going to write down your own code to compute the probability of visiting all possible states when the number of transition time increases. Simulate how the transition probability for visiting each state changes with the number of steps using the following 3 test one-step transition matrices (simulate up to 50 time steps) case 1: 0.1 0.2 0.3 0.4 0.4 0.3 0.2 0.1 0.5 0. 0.1 0.3' 0.25 0.25 0.25 0.25 case 2: 0.1 0.2 0.3 0.4 0.5 0.1 0.1 0.3 0.25 0.25 0.25 0.25 case 3: 0 0.5 0 0.5 0.5 0 0.5 0 0 0.4 0 0.6 0.3 0 0.7 0 (2) Can you find more interesting testing cases? Test those cases using your code and report your discover. (Extra Credit)

Explanation / Answer

3) Probability that he will make the goal if he missed the last shot) = P( he will make the shot given he made the ssecond last shot and missed the last one) + P(he will make the shot given he missed last two shots )

= 0.75+0.4/2 = 0.75+0.2 = 0.95

1) P(Y_n+1) = P(Y_n)*P(Y_n-1)

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