DATA ANALYSIS ENGINEERING

Question

The prevalence of a certain type of cancer among women aged 50–60 is 1 in 150. A blood

test will be positive 98% of the time if the cancer is present, but is also positive 7% of the

time if the cancer is not present. 1 In a routine check-up, a 55-year-old woman receives a

positive blood test. What is the probability that she has this type of cancer? 2 How would

you interpret this calculation if the test were ordered because of the presence of other

symptoms associated with the disease?

Answer 1

Ok, so the question is asking what the probability of the lady having the cancer is, given that she had a positive blood test. So this is a conditional probability problem, which looks something like this:

P(Cancer|Positive Test)=P(Positive Test|Cancer)*P(Cancer)/P(Positive Test)

So we know that the probability that one receives a positive test given the cancer is present is .98, and we know that the probability of having cancer is 1/150, or 0.00667. Last but not least, we have to find the marginal probability that someone has a positive test.

To get that marginal, we have to look at them testing positive under both circumstances, so we really want:

P(Positive Test & Cancer) + P(Positive Test & No Cancer). That first equation can be broken down to:

P(Positive Test|Cancer)*P(Cancer), and the second can be written has P(Positive Test|No Cancer)*P(No Cancer).

The first part of the denominator is .98*1/150, and the second part of the denominator is .07*149/150.

So the answer is: (.98*1/150)/(.98*1/150 + .07*149/150). I leave it up to you to do the actual calculations. I hope this helped you.