# Contingency and the intersection of two events.

#### Kruisman

Hi everybody,

Here is my problem:

I am being asked to solve P(R|C). C=number of patients who have completed treatment, and R=number of return patients. However, my understanding is that I need to know the intersection of P(C) and P(R) in order to solve P(R|C). But, to solve for the intersection, I need to know P(R|C). Is there a way to determine one without knowing the other?

 Doctor Number Patients Treated Treatment Completed Return Patients P(Completed) P(Return) P(R|C) Dr. Jim 2541 2134 704 0.8398268398 0.2770562771 ?

#### johng

For any events $R$ and $C$, $$P(R|C)={P(R)\over P(R\cap C)}\text{ here of course }P(R\cap C)\neq 0$$
So the computation of the conditional probability requires knowing the probability of the intersection of the events.
For your problem, with the stated probabilities, it must be true that $P(R\cup C)=1$; that is, the union of R and C is the entire population space. Now
$$P(R\cup C)=P(R)+P(C)-P(R\cap C)\text{ or }P(R\cap C)=P(R)+P(C)-1$$
I'll let you do the arithmetic.

#### Kruisman

Thank you johng. This was a big help. I thought of this option, but was unable to confirm that it was a good one. Thanks again!!