CHEN_SENG 430 - HW 4 Sol
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Texas A&M University *
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Course
430
Subject
Industrial Engineering
Date
Dec 6, 2023
Type
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5
Uploaded by BrigadierOwlPerson557
CHEN/SENG 430
HW 4 solution
1.
Consider the heart disease medical example. Let the prior probability of a heart disease, P(H) = 10
-3
,
which is the same as in the original example. Let E = a positive test, which is 98.6% reliable for those
who have the disease. Also, the test is 97.7% reliability for those who do not have the disease.
a.
Using the Bayes model, state the expression and calculate the prior predictive probability of the
observed event of a positive test, P(E).
b.
State the Bayes model expression in terms of H, E and calculate the posterior
probability of disease given a positive test, P(H|E)
c.
Use the AgenaRisk software to set up the Bayes expression in a 2-node (Boolean) Bayesian network
with H (disease) and E (Test positive) as the nodes and a directed arc or link from H to E. Enter the NPT
(node probability table) data for H and E. Run the calculation to obtain the marginal probabilities for H
and E. Within your homework paper, include a screen capture of your BN calculation shown in a graph.
d.
Enter the observation “True” for E, instantiation for E, and recalculate the BN to obtain the posterior
values of the H distribution of two values. Compare the probability of disease, H = “True”, with the
probability of disease value calculated from the Bayes expression in d). State whether these two values
agree numerically. Within your homework paper, include a screen capture of your BN calculation.
2.
Working as a sports analyst, you are modeling the free throw successes of Lebron James. You are asked
to predict the probability that Lebron will make four free throws in serial succession during an
important game that takes place in the opposite team’s venue. You express the probability of success
in the first free throw as P(T1), success in the second free throw as P(T2), and so on.
Based on your data of Lebron’s previous free throws, you estimate that the median probability (0.5
below and 0.5 above) of a single free throw P(T1) by Lebron under a wide range of expected conditions
is 0.75. Also, you estimate that the probability of success in free throw 2 given he made the free throw
1 is P(T2|T1) = 0.8. Also, you estimate that the probability of free throw 3 given he made free throw 2
(with kudos) is P(T3|T2) = 0.9. But because of the extreme conditions at the important game in the
opposite team’s site, you estimate that his probability of making free throw 4 given he was successful
in free throw 3 is 0.7.
a.
Applying the chain rule to a system of four variables, T4, T3, T2, T1, expand the joint
distribution function for the system of four variables, P(T4,T3,T2,T1). [For convenience in the
following problems, begin with variable T4, and continue with T3, then T2, and finally with T1,
so the first conditional probability is P(4|1,2,3)].
P(T1,T2,T3,T4)=P(T4|T3,T2,T1)P(T3|T2,T1)P(T2|T1)P(T1)
b.
Assuming the Markov Rule for which the probability of the variable is considered dependent
only on the previous variable, simplify the expanded expression you found in a. Explain your
answer.
P(T1,T2,T3,T4)=P(T4|T3)P(T3|T2)P(T2|T1)P(T1).
Because of the Markov Rule, variable T4 is considered dependent on the previous variable T3 but
independent of variables T2 and T1. Likewise, Variable T3 is considered dependent only on T2 but
independent of variable t1. Since each of throws are only dependent on the previous throw, any
throw before the last are not considered.
c.
Using the expression and the data, calculate Lebron’s probability of success of four free throws
in succession.
P(T1, T2,T3,T4)=P(T4|T3)P(T3|T2)P(T2|T1)P(T1)= 0.7(0.9)(0.8)(0.75)=0.38
d.
Also, calculate the mean value and variance, and standard deviation of the probabilities where
each probability value is equally weighted.
e.
Further simplify the expression you obtained in b. to predict the probability of Lebron’s making
4 free throws in series if each free throw is considered
independent
of all of the other free
throw attempts. Calculate the probability of 4 successes if all free throws are independent of
each other (Use Lebron’s median Pr value of success = 0.75).
P(T1, T2,T3,T4) = P(T1)
4
= (0.75)
4
=0.32
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f.
Compare the results obtained in c. and e. and comment on which one you think is more
realistic.
Using estimated conditional probabilities is far more realistic than assuming independence of
conditions by using data or estimates for free throws under the influences of prevailing
conditions, so the analysis including this information, even if resulting in rough estimates
given the conditions, can yield more realistic analyses and more accurate predictions than
assuming independence of the model variables.
g.
Show that the answer you obtained in e. can also be obtained if a Binomial expression was used
to calculate the probability of Lebron making the 4 free throws in succession.
P(T1,T1,T1,T1)=
�
𝟒𝟒
𝟒𝟒
� 𝟎𝟎
.
𝟕𝟕𝟕𝟕
𝟒𝟒
(
𝟏𝟏 − 𝟎𝟎
.
𝟕𝟕𝟕𝟕
)
𝟒𝟒−𝟒𝟒
=
𝟎𝟎
.
𝟑𝟑𝟑𝟑
3.