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 )

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.

ca = ec

cpctc makes it so that corresponding parts of the triangle you proved to be equal are equal, the other options aren't part of the part of the triangles you proved to be equal.