CISC 7700X Midterm Exam

1. c
2. c
3. d, see https://en.wikipedia.org/wiki/Raven_paradox
4. d
5. a
6. e, what would be the deathtoll if nobody was vaccinated?
7. c
8. b
9. b
10. d
11. e, the P(x|y)= P(x,y)/P(y) is how conditional probability is defined.
12. a
13. b
14. a
15. d, P(6s|see4) = P(see4|6s)P(6s) / ( P(see4|6s)P(6s)+P(see4|8s)P(8s))
                  = 1/6 * 1/2 / ( (1/6*1/2) + (1/8*1/2) ) 
                  = 1/12 / ( 1/12 + 1/16 ) = 4/7 
16. d
17. b
18. 0.8571
  given: P(G) = 0.5, P(-G) = 0.5, 
        P(r&d|G) = 0.6, P(-r&d|G) = 0.4
        P(r&d|-G) = 0.1, P(-r&d|-G) = 0.9
    Bayes: P(G|r&d) = P(r&d|G)P(G)/P(r&d)
                    = P(r&d|G)P(G) / ( P(r&d|G)P(G) + P(r&d|-G)P(-G) )
                    = (0.6 * 0.5) / ( 0.6 * 0.5 + 0.1 * 0.5) = 0.8571
19. Not enough information to solve the problem.
  given: P(rpe|G) = 0.8, P(-rpe|G) = 0.2
         P(rpe|-G) = 0.15, P(-rpe|-G) = 0.85
    Bayes: P(G|r&d,rpe) = P(r&d,rpe|G)P(G) / P(r&d,rpe) 
    we do not have P(r&d,rpe|G) nor P(r&d,rpe), and they may not be independent.

20. 0.9697
    We naively assume P(r&d,rpe|G) = P(r&d|G)*P(rpe|G), we can now solve: 
      P(G|r&d,rpe) = P(r&d,rpe|G)P(G) / P(r&d,rpe) 
                   = P(r&d|G)*P(rpe|G)*P(G) / (P(r&d|G)*P(rpe|G)*P(G)+P(r&d|-G)*P(rpe|-G)*P(-G))
                   = (0.6*0.8*0.5) / (0.6*0.8*0.5+0.1*0.15*0.5) = 0.9697