How to prove $P(A \cap \sim B) = P(A) − P(A \cap B)$ using the three rules of probability?
I understand this logic when drawn on a Venn Diagram, but I am unsure how it translates to a formal proof.
Thanks
2026-03-28 22:08:16.1774735696
How to prove $P(A \cap \sim B) = P(A) − P(A \cap B)$ using the three rules of probability?
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Since $A\cap B^c$ and $A\cap B$ are disjoint the probability of the union is equal to the sum of probabilities, so:
$P((A\cap B^c)\cup(A\cap B))=P(A\cap B^c)+P(A\cap B)$
but the first term on the right is equal to $P(A)$, then: $P(A)-P(A\cap B) = P(A\cap B^c)$