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 ?
I have tried it, Please correct me if I am doing something wrong or there is any other easy way of doing that.
p(A) be the probability population has cancer = 0.22 p(A`) be the probability population don't have cancer = 0.78
E1 be the event of probability that test is positive = 0.70 E2 be the event of probability that test is negative = 0.30
p(person has cancer) = p(E1|A)*E1 / [p(E1|A)*E1] * [p(E2|A`)*E2]
p(person has cancer) = 0.22*0.70/0.22*0.70+0.78*0.30
is this correct ?? or I am doing something wrong ?
Thanks . Help is appreciated