Conditional probability from a sample

Question

I would like to calculate the conditional probability $P(A|B)$ from the following sample of deaths from a certain year: \begin{array}{c|c|c} \hline & (B) \text{Smoker} & (C)\text{Not smoker} \\ \hline (A) \text{Tumor} & 231 & 324\\ \hline (D) \text{Not Tumor} & 371 & 717 \\ \hline \end{array}

I know that $P(A|B)=\frac{P(A\cap B)}{P(B)}$, and the answer is $P(A|B)=0.3837$, but I cannot get this number. I appreciate any help. Thank you.

Answer 1

You could just blindly use the formula, but also you could go by the actual meaning of it, where 'conditioning' is a fancy way of saying- "restricting our sample/possibility space".

So $A|B = $ Tumor $|$ Smoker = "Tumorous patients restricting our attention to the sample space of only Smokers".

The sample space of only smokers consists of $231 + 371 = 602$ people. This is our new sample space. Of this sample space, $231$ are $A$ and the rest are $D$. So using basic probability theory- the probability of something happening is the 'size it takes up' within the sample space, i.e, $231/602 = 0.3837$.