Question

Give a mathematical derivation of the formula P((A ∩ Bc ) ∪ (Ac ∩ B)) = P(A) + P(B) − 2P(A ∩ B). Your derivation should be a sequence of steps, with each step justified by appealing to one of the probability axioms.

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1 Since the events A ∩ Bc and Ac ∩ B are disjoint, we have, using the additivity axiom, P((A ∩ Bc ) ∪ (Ac ∩ B)) = P(A ∩ Bc ) + P(Ac ∩ B).

Since A = (A∩B)∪(A∩Bc) is the union of two disjoint sets, we have, again by the additivity axiom,

P(A) = P(A ∩ B) + P(A ∩ Bc ), so that

P(A ∩ Bc ) = P(A) − P(A ∩ B). similarly

P(B ∩ Ac ) = P(B) − P(A ∩ B).

Therefore, P(A ∩ Bc ) + P(Ac ∩ B) = P(A) − P(A ∩ B) + P(B) − P(A ∩ B) = P(A) + P(B) − 2P(A ∩ B).

#####q1#####

1-how did they conclude (A ∩ Bc ) and (Ac ∩ B) are disjoint events?

2-how is P(A ∩ Bc ) = P(A) − P(A ∩ B) ?

#####q2#####

2-Consider a probabilistic model whose sample space is the real line. Show that P([0, ∞)) = lim P([0, n]) and lim P([n, ∞)) = 0.

Answer 1