Venn Diagram Probabilities

Question

We see that $P(A^c\cap C ) = P(C) - P(A\cap C)$.

How do we derive from $P(A^c\cap C )$ to $P(C) - P(A\cap C)$ without looking at a diagram?

$P(A^c\cap C ) = P(A^c)P(C|A^c)$

= ... ?

= $P(C) - P(A\cap C)$

Answer 1

This becomes immediate if you relativize the probabilities. Every event falls within either $A$ or $A^c$, so $P(A)+P(A^c)=P(U)$. Intersect each of these sets with $C$ to get

$$P(A\cap C)+P(A^c\cap C)=P(U\cap C)=P(C)$$

Answer 2

$$C = AC \cup A^cC$$

Notice that $AC$ and $A^cC$ are disjoint.

$$P(C)=P(AC)+P(A^cC)-P(AC \cap A^cC)=P(AC)+P(A^cC)-0$$

$$P(A^cC)=P(C)-P(AC)$$