Question

The probability that a randomly selected person has cancer in 0.02. The probability that he or she reacts positively to a test which detects cancer is 0.95 if he or she has cancer and 0.03 if he or she does not. Determine the probability that a randomly tested person:
a) reacts positively
b) has cancer given that he or she reacts positively

Answer 1

I don't understand why people who have cancer react more positively than than those who don't.

Answer 2

\(C\) = Has cancer
\(P\) = Reacts positively

We know \(\Pr(C) = 0.02\), \(\Pr(P|C)=0.95\), and \(\Pr(P|C^C) = 0.03\).

Answer 3

They want \(\Pr(P)\) for (a) and \(Pr(C|P)\) for (b).

Answer 4

For (a) we use law of total probability: \(\Pr(P) = \Pr(P|C\cup P|C^C)=\Pr(P|C) + \Pr(P|C^C)\)

Answer 5

For (b) we use Bayes' theorem: \[
\Pr(C|P) = \frac{\Pr(P|C)\Pr(C)}{\Pr(P)}
\]