Question

Suppose A and B are two independent events with probabilities $P\left(A\right)\ne 0\text{and}P\left(B\right)\ne 0.$ $\text{Let}\overline{A}\text{and}\overline{B}$ be their complements. Which one of the following statements is FALSE?

• A. $P(A\cap B)=P\left(A\right)P\left(B\right)$
• B. $P\left(A/B\right)=P\left(A\right)$
• C. $P(A\cup B)=P\left(A\right)+P\left(B\right)$
• D. $P(\overline{A}\cap \overline{B})=P\left(\overline{A}\right).P\left(\overline{B}\right)$

Answer 1

The correct option is C $P(A\cup B)=P\left(A\right)+P\left(B\right)$

If A and B are independent events then
$P(A\cap B)=P\left(A\right)P\left(B\right)$

$P\left(A/B\right)=\frac{P(A\cap B)}{P\left(B\right)}=\frac{P\left(A\right)P\left(B\right)}{P\left(B\right)}=P\left(A\right)$

If A and B are independent events then $P\left(\overline{A}\right)\text{and}P\left(\overline{B}\right)$ are also independent.

& $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$