the event that A or B but not both will occur can be written as
(A∩ B')∪(A' ∩ B)
express the probability of this event in terms of P(A), P(B), and P(A∩ B)
please step by step
=[(A∩ B')∪A'] ∩ [(A∩ B')∪B]
=[(A∪A')∩( B'∪A')]∩ [(A∪B)∩( B'∪B)]
=( B'∪A')∩(A∪B)
=(A ∩B)' ∩(A∪B)
=(A∪B)-(A∩B)
∴P[(A∩ B')∪(A' ∩ B)]
=P[(A∪B)-(A∩B)]
=P(A∪B)-P(A∩B) (as A∩B⊂A∪B)
=P(A)+P(B)-P(A∩B)-P(A∩B)
=P(A)+P(B)-2P(A∩B)
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Verified answer
(A∩ B')∪(A' ∩ B)
=[(A∩ B')∪A'] ∩ [(A∩ B')∪B]
=[(A∪A')∩( B'∪A')]∩ [(A∪B)∩( B'∪B)]
=( B'∪A')∩(A∪B)
=(A ∩B)' ∩(A∪B)
=(A∪B)-(A∩B)
∴P[(A∩ B')∪(A' ∩ B)]
=P[(A∪B)-(A∩B)]
=P(A∪B)-P(A∩B) (as A∩B⊂A∪B)
=P(A)+P(B)-P(A∩B)-P(A∩B)
=P(A)+P(B)-2P(A∩B)