Question

If $A,B,C$ are three events show that
$P(A\cup B\cup C)=P\left(A\right)+P\left(B\right)+P\left(C\right)-P(A\cap B)-P(b\cap C)-P(C\cap A)+P(A\cap B\cap C)$

Answer 1

M= A ∪ B N = C

P(A∪B∪C)= P(M∪N)

= P(M) + P(N) - P(M$\cap$N)

= P(A∪B) + P(C) - P((A ∪ B)$\cap$C)

= P(A∪B) + P(C) - P(AC ∪ B$\cap$C)

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

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

= P(A) + P(B) + P(C) - P(A$\cap$B) - P(A$\cap$C) - P(B$\cap$C) + P(A$\cap$B$\cap$C) q.e.d.