Question

Find P(C|Y) from the information in the table. To the nearest tenth, P(C|Y) =

Answer 1

From the definition of conditional probability,

$\mathbb P(C\mid Y)=\dfrac{\mathbb P(C\cap Y)}{\mathbb P(Y)}$

and by the law of total probability,

$\mathbb P(Y)=\mathbb P(Y\cap A)+\mathbb P(Y\cap B)+\mathbb P(Y\cap C)$

In other words, each cell in the grid left of and above the total column and row represents the joint probabilities (e.g. $\mathbb P(X\cap A)=\dfrac{32}{146}$). So,

$\mathbb P(C\mid Y)=\dfrac{15}{30}=\dfrac12=0.5$