# maximum conditional probability

A company offers a test in which 95% of postive tested people are actually postive. The company hides that 15% of non-postive people are tested postive. You assume maximal 3 out of 15 players in your local club are postive ( e.g. doping).

a) Can you belive the Data given by the company? No, because total probability gives 0.95*3/15+0.15*12/15=4.65 > 3 so either the assumption or test is wrong.

b) Assume, the probability of being tested postive when postive is 0.95. How likely is it maximal that a postive tested player ist actually postive. $P(A|B)= \frac{P(A intersect B)}{P(B)}$ A=postive, B=postive tested. P(B)=0.31 , what ist meant by maximal? I thought P(A intersect B)=0.95 but then the conditional probability is greater than 1? can someone help me here?

c) Assume the Data given by the company are right, do you have to increase your assumption of postive players in your team? Either I dont get the question or it's a trivial yes. Can someone explain this to me?

Thank you.

You should use Bayes theorem in part (b). Let $p(T_P)$ be the probability of testing positive. Let $p(P)$ and $p(NP)$ be the probabilities of "being positive" and "not being positive", respectively. You have used, in your calculations in part (a), that $p(P)=3/15$ and $p(NP)=12/15$. Now it is given in part (b) that $p(T_P|P) = 0.95$ and you have to calculate $p(P|T_P)$. Using Bayes theorem
$p(P|T_P) = \frac{p(T_P|P)p(P)}{p(T_p)}$. Also using total probability $p(T_p) = p(T_p|P)p(P) + p(T_p|NP)p(NP)$, where $p(T_p|NP) = 0.15$ as provided in the statement of the question.