SOLUTION: P% of the country's population have symptoms of a certain disease. When tested, a person with symptoms is diagnosed with a probability of p1, while a person without symptoms tests

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Question 1200744: P% of the country's population have symptoms of a certain disease. When tested, a person with symptoms is diagnosed with a probability of p1, while a person without symptoms tests positive with a probability of p2. What is the probability that a randomly tested person will have a positive test result? What is the probability that a person who tested positive actually has symptoms of the disease? ([P, p1, p2] = [14, 0.96, 0.15])
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
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P% of the country's population have symptoms of a certain disease.
When tested, a person with symptoms is diagnosed with a probability of p1,
while a person without symptoms tests positive with a probability of p2.
(a) What is the probability that a randomly tested person will have a positive test result?
(b) What is the probability that a person who tested positive actually has symptoms of the disease?
([P, p1, p2] = [14, 0.96, 0.15])
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(a)  The logic of the solution is as follows:

         We have two disjoint sets: one "with the symptomps" and the other "without the symptoms".

         If a person belongs to the first set, he contributes to the sough probability 
         with the weight p1; 
         if a person belongs to the second set, he contributes to the sough probability 
         with the weight p2.


         It gives the ANSWER to question (a)  0.14*0.96 + (1-0.14)*0.15 = 0.2634.



(b)  In the second question, they want you calculate a conditional probability.

     So, this probability is the ratio of two values.
     The denominator is the probability that a randomly chosen person is tested positively.

     It is the probability which we just found out in part (a) above.


     The numerator is the measure of the intersection of two sub-sets: those who has symptoms AND, 
     at the same time, are tested positively. So, the probability in the numerator is

            0.14*0.96.


      Thus the final expression and the value for this conditional probability is

            %280.14%2A0.96%29%2F%280.14%2A0.96+%2B+%281-0.14%29%2A0.15%29 = 0.51025   (rounded).    ANSWER

Solved.