Question

Which of the following is not necessarily true for independent events A and B? P (A intersection B )equals P (A )P (B )P (A vertical line B )equals P (A )P (B vertical line A )equals P (B )P (A U B )equals P (A )plus P (B )

Answer 1

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.