Answer:
P(A U B)=P(A)+P(B)
Step-by-step explanation:
Reading the options that we have for the answer, one of them (the first one) is the definition of being independent. A and B are independent if and only if P(A ∩ B)=P(A)*P(B).
So the first one IS necessary true for independent events and with this equation, option two and three are necessary true for independent events:
For definition of P(A | B)
P(A | B)= P(A ∩ B) / P(B)
And we replace P(A ∩ B) using the first option that we know it´s true:
P(A | B)= P(A)*P(B) / P(B)= P(A)
So P(A | B)=P(A) it´s true for A and B independent.
The same process goes to show P(B | A)=P(B)
Because of this, the only one of the options that could not be true for independent events is P(A ∪ B)=P(A) + P(B), and this happens because P(A ∩ B)=P(A)*P(B) applies but it could be different from 0 considering P(A ∪ B)=P(A) + P(B) - P(A ∩ B). We conclude this property (P(A ∪ B)=P(A) + P(B)) is not necessary true for A and B independent.
