SOLUTION: A certain disease has an incidence rate of 0.5%. The false-negative rate on a test for the disease is 7%; the false positive rate is 5%. Compute the probability that a person who t

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Question 1191926: A certain disease has an incidence rate of 0.5%. The false-negative rate on a test for the disease is 7%; the false positive rate is 5%. Compute the probability that a person who tests positive actually has the disease. (You may find it useful to construct a probability contingency table.)
The probability a person who tests positive actually has the disease is
Give your answer accurate to at least 3 decimal places
Answer by ikleyn(52810) About Me  (Show Source):
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A certain disease has an incidence rate of 0.5%.
The false-negative rate on a test for the disease is 7%;
the false positive rate is 5%.
Compute the probability that a person who tests positive actually has the disease.
(You may find it useful to construct a probability contingency table.)
The probability a person who tests positive actually has the disease is
Give your answer accurate to at least 3 decimal places
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            This problem is about calculating the  CONDITIONAL  PROBABILITY.

            They want you calculate this ratio   tests_positive_AND_has_the_disease%2Ftests_positive. 


Let X be the total population.


Then the number of those who has the disease is 0.005*X;  the number of those who has no the disease is 0.995*X.


     The number of those who has    the disease and tests positive is  0.005*(1-0.07)X = 0.005*0.93*X.

         To calculate this amount, I excluded from 0.005X, the number of sick persons, the value 0.005X*0.07, 
         which represents the part of sick who are tested as healthy (i.e. are false-negative).


     The number of those who has no the disease and (or "but") tests positive is  0.995*0.05*X              (false test positive).


So the number of those who tests positive is the sum  0.005*0.93*X + 0.995*0.05*X.


They want you calculate  the conditional probability


    P = tests_positive_AND_has_the_disease%2Ftests_positive = %280.005%2A0.93%2AX%29%2F%280.005%2A0.93%2AX+%2B+0.995%2A0.05%2AX%29 = cancel X in the numerator and in the denominator = 

                   = %280.005%2A0.93%29%2F%280.005%2A0.93+%2B+0.995%2A0.05%29 = 0.00465%2F%280.00465%2B0.04975%29 = use your calculator = 0.085478  (rounded).    ANSWER


ANSWER.  This probability is  0.085478.

Solved.