Cancer is present in 22% of a population and is not present in the remaining 78%. An imperfect clinical test successfully detects the disease and with probability 0.70. Thus, if a person has the disease in the serious form, the probability is 0.70 that the test will be positive and it is 0.30 if the test is negative. Moreover among the unaffected persons, the probability that the test will be positive is 0.05. A person selected at random from the population is given the test and the result is positive. What is the probability that this person has the cancer ?
p(A) = Person Has Cancer p(X) = Positive Result
p(A|X) => p(X|A)P(A)/p(X|A)P(A)+p(X|~A)P(~A)
p(A) = 0.70 p(~A) = 0.30 p(X|A) = 22% = 0.22 p(X|~A) = 0.78
=> 0.22 * 0.70/0.22*0.70 + 0.78*0.30 => 0.397
Is my answer and concept is right towards this question ? if not, please correct me out. I am new to probability and help is appreciated .