Question 1189105
.
A test to detect cancer is not always reliable. 
It gives a positive result 95 % of the time if the person does have cancer, 
and it gives a positive result 3 % of the time that the person does not. 
The probability that a randomly selected person has cancer is 0.02.
a. Given that a test on a randomly selected person is positive, 
find the probability that he/she does have cancer? Ans: 0.393
b. Out of 5000 people on which this test is used, how many people with cancer 
would you expect to be correctly diagnosed? Ans: 95 people
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<U>PART (a)</U>



<pre>
Consider a population of 100,000  people.


Of them, 0.02 have cancer, i.e. 0.02*100,000 = 2000, according to the problem.


Of these 2000, 95% will have a true positive test, i.e. 1900 persons.


Of the remaining 100,000-2,000 = 98,000 people, 3% will have a false positive test, i.e. 2940 persons.


In all, of 100,000 population, 1900 + 2940 = 4840 have positive cancer test.


The ratio  {{{have_cancer/have_positive_cancer_test}}} = {{{1900/4840}}} = 0.393,  rounded.


It is the answer to question (a), deduced informally.


Formally, it is  P = {{{(0.02*0.95)/(0.02*0.95 + (1-0.02)*0.03)}}} = 0.393  (rounded).
</pre>