Can someone explain why this proof is incorrect? In addition, what must we
assume for this to be correct?
Let S be the sample space of the experiment and let E be any event in S.
Denote the number of elements in an event E by n(E) and let the number of
elements in S be N, then
P(E) = n(E)/N
Let X = A intersect B'
Y = A intersect B
Z = A' intersect B. Then
A union B = (A intersect B') union (A intersect B) union (A' intersect B)
and the events
(A intersect B') , (A intersect B) and (A' intersect B) are
disjoint. hence,
P(A union B) = n(A union B)/N
= (n(X) + n(Y) + n(Z)) / N
= [ [n(X) + n(Y)] + [n(Y) + n(Z)] - n(Y) ] / N
= [ n(A) + n(B) -n(A intersect B) ] / N
= n(A)/N + n(B)/N + n(A intersect B)/N
= P(A) + P(B) - P(A intersect B)
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