From the definition of conditional probability,
[tex]\mathbb P(C\mid Y)=\dfrac{\mathbb P(C\cap Y)}{\mathbb P(Y)}[/tex]
and by the law of total probability,
[tex]\mathbb P(Y)=\mathbb P(Y\cap A)+\mathbb P(Y\cap B)+\mathbb P(Y\cap C)[/tex]
In other words, each cell in the grid left of and above the total column and row represents the joint probabilities (e.g. [tex]\mathbb P(X\cap A)=\dfrac{32}{146}[/tex]). So,
[tex]\mathbb P(C\mid Y)=\dfrac{15}{30}=\dfrac12=0.5[/tex]
