CISC 7700X Midterm Exam

1. c
2. d
3. b
4. b
5. b
6. for(f in all_values_of_F){
    out[f] = 0
   }
   for(a in all_values_of_A){
    for(b in all_values_of_B){
     for(c in all_values_of_C){
      for(d in all_values_of_D){
       for(e in all_values_of_E){
        for(f in all_values_of_F){
         out[f] += P(a,b,c,d,e,f)
        }
       }
      }
     }
    }
   }

7. 0.75

  all outcomes:
     0 wins: 0.125, e.g. $1 * 0.5 * 0.5 * 0.5
     1 wins: 0.375,0.375,0.375
     2 wins: 1.125,1.125,1.125
     3 wins: 3.375
  median is (0.375+1.125)/2 = 0.75

8. 0.649519

  eg. exp((log(0.125)+3*log(0.375)+3*log(1.125)+log(3.375))/8)

9. 1.

  all outcomes:
     0 wins: 0.125, e.g. $1 * 0.5 * 0.5 * 0.5
     1 wins: 0.375,0.375,0.375
     2 wins: 1.125,1.125,1.125
     3 wins: 3.375
  mean is (0.125 + 3*0.375 + 3*1.125 + 3.375)/8 = 1

10. b
11. c
12. d
13. c
14. e, identify, that is how conditional probability is defined.
15. 0.2
 P(Return) = 0.1, P(-Return) = 0.9
 P(Sale) = 0.3, P(-Sale) = 0.7
 P(Sale|Return) = 0.6, P(-Sale|Return) = 0.4

 P(Return|Sale) = P(Sale|Return)P(Return) / P(Sale) 
                = 0.6 * 0.1 / 0.3 = 0.2
16. 0.5
 P(Fragile|Return) = 0.9, P(-Fragile|Return) = 0.1
 P(Fragile|-Return) = 0.1, P(-Fragile|-Return) = 0.9

 P(Return|Fragile) = P(Fragile|Return)*P(Return) / P(Fragile)
 
 # marginalization:
 P(Fragile) = P(Fragile|Return)*P(Return) + P(Fragile|-Return)*P(-Return)
  P(Fragile) = 0.9*0.1 + 0.1*0.9 = 0.1800

 P(Return|Fragile) = P(Fragile|Return)*P(Return) / P(Fragile)
 P(Return|Fragile) =  (0.9*0.1) / 0.1800 = 0.5

17. No answer.
 
 We need: P(Return|Sale,Fragile) = 
     P(Sale,Fragile|Return)*P(Return)/P(Sale,Fragile).
 We are not given P(Sale,Fragile) nor are we told that Sale and Fragile
 are independent (e.g. what if the store puts fragile items on sale often?).

18. 0.6923

  P(Return|Sale,Fragile) = P(Sale,Fragile|Return)*P(Return)/P(Sale,Fragile).
  we *assume* that Sale and Fragile are independent:
   e.g. P(Sale,Fragile) = P(Sale)*P(Fragile)

  Naive bayes is: 
   P(Return|Sale,Fragile) = 
     P(Sale|Return)*P(Fragile|Return)*P(Return) / 
      (P(Sale|Return)*P(Fragile|Return)*P(Return)+P(Sale|-Return)*P(Fragile|-Return)*P(-Return))

  From previous questions, we have everything except: P(Sale|-Return)
   We know P(Sale) = P(Sale|Return)*P(Return)+P(Sale|-Return)*P(-Return)
                0.3 = 0.6*0.1 + P(Sale|-Return)*0.9
                P(Sale|-Return) = (0.3 - 0.6*0.1)/0.9 = 0.2667

    P(Return|Sale,Fragile) ~ (0.6*0.9*0.1) / (0.6*0.9*0.1 + 0.2667*0.1*0.9) = 0.6923

19. We're comparing different sized populations, 100k vs 900k. 
  e.g.: 
  If everyone (1,000,000) drank regular coke, then we would see 5000 hospitalizations, vs
  if everyone (1,000,000) drank diet code, we would only see 250 hospitalizations.

20. a