If A and B are two events, then P $(A \cap B)$ = _____________.

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Solution

For two events A and B,
P $(A \cap B)$ = ??
Using Venn-diagram

$Clearly (A \cap B) \cup (A \cap B) = B ∴ P (B) = P ((A \cap B) \cup (A \cap B)) = P (A \cap B) + P (A \cap B) - P ((A \cap B) \cap (A \cap B)) = P (A \cap B) + P (A \cap B) - P (A \cap A \cap B) [Intersection commutes] = P (A \cap B) + P (A \cap B) - P (ϕ) [∵ A \cap A = ϕ] i . e P (B) = P (A \cap B) + P (A \cap B) - 0 ∴ P (A \cap B) = P (B) - P (A \cap B)$

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A

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