Question

Let A and B be any two events. Which of the following statements, in general, are false?

a. P(AB) + P(Aˉ|Bˉ) = 1.
b. P(AB) + P(A|Bˉ) = 1.
c. P(AB) + P(Aˉ|B) = 1.

Answer 1

In general, option c is false because P(AB) + P(Aˉ|B) does not equal 1.

Step-by-step explanation:

In general, the statement that is false is c. P(AB) + P(Aˉ|B) = 1.

Using the probability definitions, we can rewrite the statements as follows:

a. P(AB) + P(Aˉ | Bˉ) = P(A)P(B) + P(A' | B') = P(A)P(B) + P(A')P(B') = P(A) + P(A')P(B') = 1, which is true.

b. P(AB) + P(A | Bˉ) = P(A)P(B) + P(A | B') = P(A)P(B) + P(A)P(B') = P(A)(P(B) + P(B')) = P(A) + P(A)P(B') = 1, which is true.

c. P(AB) + P(Aˉ | B) = P(A)P(B) + P(A' | B) = P(A)P(B) + P(A')P(B) = P(A)(P(B) + P(B)) = P(A) + P(A)P(B) = P(A)(1 + P(B)) ≠ 1, which is false.

Therefore, option c is generally false.