Question

Consider the following statements:
(1) $P(\overline{A}\cup B)=P\left(\overline{A}\right)+P\left(B\right)-P(\overline{A}\cap B)$
(2) $P(A\cap \overline{B})=P\left(\overline{B}\right)-P(A\cap B)$
(3) $P(A\cap B)=P\left(B\right)P\left(A|B\right)$
Which of the above statements are correct?

• A. 1 and 2 only
• B. 1 and 3 only
• C. 2 and 3 only
• D. 1, 2 and 3

Answer 1

The correct option is B 1 and 3 only

Statement 1: $P(\overline{A}\cup B)=P\left(\overline{A}\right)+P\left(B\right)-P(\overline{A}\cap B)$ is true by definition.

Statement 2: $P(A\cup \overline{B})=P\left(B\right)-P(A\cap B)$ is false by definition.

Statement 3: $P(A\cap B)=P\left(B\right)P\left(A|B\right)$ is true by definition of conditional probability.

Hence, only statements 1 and 3 are true.