1. Chain Rule and the Markov Rule

1. You as a sports analyst are modeling the free throw success ofLebron James. You are asked to predict the probability that Lebronwill make four free throws in serial succession during an importantgame that takes place in the opposite team’s venue.

Based on your data ofLebron’s previous free throws, you estimate that the medianprobability (0.5 below and 0.5 above) of a single free throw byLebron under a wide range of expected conditions is 0.75. Also,estimate that the probability of free throw 2 given he made freethrow 1 is P(2|1) = 0.8. Also you estimate that the Pr of freethrow 3 given he made free throw 2 with kudos is P(3|2) = 0.9. Butbecause of the extreme conditions at the important game in theopposite team’s site, you estimate that the Pr he makes free throw4 given he made free throw 3 is 0.7.

Using these data, write the expression for and predict to 2 sdthe Pr of success of four free throws in succession. Assume thatthe Pr he makes the first free throw is his median value of 0.75.As in b) assume the Markov Rule in which the primary dependency iswith the previous free throw and further back free throws are lessinfluential and considered independent for this analysis. Also,calculate the mean value and variance, and standard deviation ofthe probabilities where each probability value is equally weighted.Finally write an expression and calculate the numerical value ofP(1,2,3,4) to two standard deviations, which represents uncertaintymanagement of estimating the epistemic uncertainty of theprobability values.

              Mean Probabilities=

Variance =

               Standard Deviation =  

P(1,2,3,4) =      to two standarddeviations, which represents uncertainty management of epistemicuncertainty in the estimated probability values.

1. b. Now predict the probability to2 sd of Lebron’s making 4 free throws in series if each free throwis considered independent of all of the other free throw attempts.Use Lebron’s median Pr value of success = 0.75.

P(1,2,3,4) =

Assume the same standard deviation calculated in a:

P(1,2,3,4)=        to two standarddeviations exhibiting an estimate of epistemic uncertainty in theanalysis. Do the results for c) and d) agree within the combinedestimated uncertainties in the two values?

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