2.7 Assume there are two urns with the following numbers of red and blue marbles
ID: 3303047 • Letter: 2
Question
2.7 Assume there are two urns with the following numbers of red and blue marbles: Um Red Blue An urn is selected (Pr[U-0.25 and Pr[U2 0.75), and then a marble is selected from that urn (assume all marbles in that urn are equally likely). Let R denote the event the marble is red and Bthe event the marble is blue. a. What are Pr[R and Pr[B]? b. What are Pr(VilR) and Pr(UIB)? c. What are PrlU3IR) and Pr[UIB)? d. Which combinations of conditional probabilities above sum to I? Why? 5. A rare disease exists with which only 1 in 500 is affected. A test for the disease exists, but of course it is not infallible. A correct positive result (patient actually has the disease) occurs 95% of the time, while a false positive result (patient does not have the disease) occurs 1% of the time. If a randomly selected individual is tested and the result is positive, what is the probability that the individual had the disease?Explanation / Answer
Ans:
2.7)a)P(R)=7/16
P(B)=9/16
b)P(U1/R)=5/7
P(U1/B)=3/9
c)P(U2/R)=2/7
P(U2/B)=6/9
d)P(U1/B)+(U2/B)=3/9+6/9=1
P(U1/R)+P(U2/R)=5/7+2/7=1
5)
P(desease)=1/500=0.002
P(not desease)=1-0.002=0.998
P(positive/desease)=0.95
P(positive/not desease)=0.01
P(desease/positive)=P(positive/desease)*P(desease)/[P(positive/desease)*P(desease)+P(positive/ not desease)*P( not desease)]
=0.95*0.002/[0.95*0.002+0.01*0.998]
=0.0019/[0.0019+0.00998]
=0.0019/0.01188
=0.1599
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