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CSE422 Assignment 1 Solution

a. Construct the joint probability distribution table for the variables Color (C) and Size (S).
b. Calculate the joint probability P(C=Red,S=Large).
c. Calculate the marginal probabilities P(C=Red) and P(S=Large).
d. Determine whether the preferences for Color and Size are independent.
e. What is the conditional probability that a respondent prefers a Small size given that they chose Red color P(S=Small∣C=Red)?
● 40 students preferred the Library in the Morning with a Quiet environment. ● 20 students preferred the Library in the Morning with Moderate noise.
● 30 students preferred the Library in the Evening with a Quiet environment.
● 10 students preferred the Library in the Evening with Moderate noise.
● 25 students preferred the Cafe in the Morning with Quiet environment.
● 15 students preferred the Cafe in the Morning with Moderate noise.
● 45 students preferred the Cafe in the Evening with a Quiet environment. ● 15 students preferred the Cafe in the Evening with Moderate noise.
Questions:
a. Create a joint probability distribution table for the variables Location (L), Time of Day (T), and Noise Level (N).
b. Calculate the joint probability P(L=Cafe,T=Evening,N=Quiet).
c. Calculate the marginal probabilities P(L=Library), and P(N=Quiet).
d. Determine if the factors Location, Time of Day, and Noise Level are independent.
Hint: Check if
P(L=Library,T=Morning,N=Quiet)=P(L=Library)P(T=Morning)P(N=Quiet).
e. What is the conditional probability that a student prefers the Library given that it is Morning and the environment is Quiet P(L=Library∣T=Morning,N=Quiet)?
4. Suppose we have three events A, B, and C within a probability space. The events A and B are known to be conditionally independent given C. This means that P(A∩B∣C)=P(A∣C)×P(B∣C). Given the following probabilities:
○ P(A∣C)=0.4
○ P(B∣C)=0.5
○ P(C)=0.2
Calculate the probability P(A∩B∩C).
5. Let A, B, and C be events in a probability space, where A and B are conditionally independent given C. Assume the following probabilities:
○ P(A∣C)=0.3
○ P(B∣^C)=0.6
○ P(C)=0.5
Calculate: P(A∩B∣C), P(A∩B∣^C) assuming A and B are conditionally independent given ^C with P(A∣^C)=0.2.
6. In a medical study, events A, B, and D represent having disease A, disease B, and taking drug D respectively. It is known that having disease A and B are conditionally independent given the use of drug D. The probabilities are given as:
○ P(A∣D)=0.4
○ P(B∣D)=0.5
○ P(D)=0.3
○ P(A∣^D)=0.2
○ P(B∣^D)=0.3
Calculate: P(A∩B∣D) and, P(A∩B∣^D) assuming A and B are also conditionally independent given ^D.
7. Consider three events E, F, and G in a probability space where E and F are conditionally independent given both G and ^G. You are given:
○ P(E∣G)=0.5
○ P(F∣G)=0.6
○ P(E∣^G)=0.4
○ P(F∣^G)=0.3
○ P(G)=0.7
Calculate: P(E∩F∣G) and, P(E∩F∣^G).
8. In a study on social media influence, researchers are trying to understand if sharing political content (Event A) and engaging in political discussions (Event B) are conditionally independent given a user’s political affiliation (Event C).
Given Probabilities: P(A∣C)=0.4, P(B∣C)=0.5, P(C)=0.3
Calculate the probability that a randomly selected user from the sample shares political content and engages in discussions, given their political affiliation.
9. A company runs two different types of marketing campaigns simultaneously: email marketing (Event D) and social media ads (Event E). They want to see if these campaigns independently attract new customers (Event F) when considering the customer segment targeted (youth, adults, etc.).
Given Probabilities: P(D∣F)=0.6, P(E∣F)=0.7, P(F)=0.2, P(D∣^F)=0.2, P(E∣^F)=0.1 Calculate the probability that a new customer is attracted by both the email marketing and social media ad campaigns, given that they are a part of the targeted segment.
Given Probabilities: P(G∣I)=0.3, P(H∣I)=0.4, P(I)=0.5.
Determine the probability that a student participates in both sports and music programs given their grade level.
11. A streaming service uses a Naive Bayes classifier to predict the genre of movies based on two features: presence of action scenes (A) and presence of scary scenes (R). The genres considered are action (X) and horror (Y).
Given Probabilities: P(X)=0.6, P(Y)=0.4, P(A∣X)=0.7, P(A∣Y)=0.3, P(R∣X)=0.2 P(R∣Y)=0.8.
Calculate the probability that a movie is a horror given that it contains action scenes and scary scenes.
12. A career counseling tool uses Naive Bayes to advise students on potential career paths based on their interest in mathematics (M) and their interest in biology (B). The career paths suggested are engineering (E) and medicine (D).
Given Probabilities: P(E)=0.7, P(D)=0.3, P(M∣E)=0.8, P(M∣D)=0.3, P(B∣E)=0.2, P(B∣D)=0.7.
What is the probability that a student is advised to pursue medicine given they have an interest in both mathematics and biology?
13. A political analyst uses a Naive Bayes classifier to predict voter behavior based on two issues: support for environmental policies (EP) and support for economic policies (EC). The classifications are progressive voter (P) and conservative voter (C).
Given Probabilities: P(P)=0.5, P(C)=0.5, P(EP∣P)=0.8, P(EP∣C)=0.3, P(EC∣P)=0.4, P(EC∣C)=0.7.
Estimate the probability that a voter is progressive given their support for both environmental and economic policies.
Given Probabilities:
○ P(Click)=0.3, P(No Click)=0.7
○ P(Y∣Click)=0.4, P(Y∣No Click)=0.2
○ P(F∣Click)=0.7, P(F∣No Click)=0.3
15. A health insurance company uses Naive Bayes classification to assess the risk of chronic illness based on smoking status (smoker S or non-smoker N) and exercise frequency (regular R or irregular I).
Given Probabilities:
○ P(High Risk)=0.25, P(Low Risk)=0.75
○ P(S∣High Risk)=0.6, P(S∣Low Risk)=0.3
○ P(R∣High Risk)=0.3, P(R∣Low Risk)=0.7
Calculate the probability that an individual is at high risk for chronic illness if they are a
smoker and do not exercise regularly.
Given Probabilities:
○ P(Party A)=0.45, P(Party B)=0.55
○ P(Y∣Party A)=0.3, P(M∣Party A)=0.4, P(O∣Party A)=0.3
○ P(L∣Party A)=0.2, P(D∣Party A)=0.5, P(H∣Party A)=0.3
○ P(HS∣Party A)=0.25, P(C∣Party A)=0.50, P(PG∣Party A)=0.25
17. A streaming service uses Naive Bayes to decide whether to show a new sci-fi series or a romantic comedy to a user, based on their previous genre preferences (sci-fi SF, romance RM), viewing time (peak PK, off-peak OP), and subscription type (basic B, premium P).
Given Probabilities:
○ P(Sci-Fi)=0.6, P(Rom-Com)=0.4
○ P(SF∣Sci-Fi)=0.7, P(RM∣Rom-Com)=0.8
○ P(PK∣Sci-Fi)=0.8, P(OP∣Rom-Com)=0.6
○ P(B∣Sci-Fi)=0.5, P(P∣Rom-Com)=0.5
Estimate the probability that a premium user, who prefers sci-fi and watches during peak times, will be shown the new sci-fi series.
18. A health app predicts whether a user is at low or high risk for diabetes based on their physical activity level (active A, sedentary S), diet type (balanced B, high-sugar HS), and family history (yes Y, no N).
Given Probabilities:
○ P(Low Risk)=0.7, P(High Risk)=0.3
○ P(A∣Low Risk)=0.8,P(S∣High Risk)=0.7
○ P(B∣Low Risk)=0.9, P(HS∣High Risk)=0.6
○ P(Y∣High Risk)=0.4, P(N∣Low Risk)=0.85
What is the probability that a sedentary user with a high-sugar diet and a family history of diabetes is at high risk?
19. A political analyst uses Naive Bayes to estimate support (Support S or Oppose O) for a candidate based on voter registration status (Registered R, Not Registered N) and past voting frequency (Frequent F, Infrequent I, Never V).
Given Probabilities:
○ P(S)=0.7, P(O)=0.3
○ P(R∣S)=0.9, P(N∣S)=0.1
○ P(F∣S)=0.6, P(I∣S)=0.3, P(V∣S)=0.1
○ P(R∣O)=0.6, P(N∣O)=0.4
○ P(F∣O)=0.2, P(I∣O)=0.5, P(V∣O)=0.3
What is the probability that a voter supports the candidate if they are registered and have never voted before?
Given Probabilities:
● P(Hire)=0.7, P(Not Hire)=0.3
● P(C∣Hire)=0.9, P(U∣Hire)=0.85
● P(C∣Not Hire)=0.4, P(U∣Not Hire)=0.3
● P(M∣Hire)=0.2, P(M∣Not Hire)=0.8
● P(N∣Hire)=0.8, P(N∣Not Hire)=0.2

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