Question

Let A and B be two events such that P(A) = 0.3, P(B) = 0.6 and $P(\frac{B}{A})$ = 0.5. Then $P(\frac{\overline{A}}{\overline{B}})$ equals

• A. $\frac{3}{4}$
• B. $\frac{5}{8}$
• C. $\frac{9}{40}$
• D. $\frac{1}{4}$

Answer 1

Answer: (B)

$\frac{5}{8}$

Answer 2

Here's how to solve this problem using conditional probability and De Morgan's law:

1. Calculate $P(A \cap B)$ using $P(\frac{B}{A}) = \frac{P(A \cap B)}{P(A)}$:

   $P(A \cap B) = P(\frac{B}{A}) \times P(A) = 0.5 \times 0.3 = 0.15$

2. Calculate $P(A \cup B)$ using:

   $P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.3 + 0.6 - 0.15 = 0.75$

3. Calculate $P(\overline{A} \cap \overline{B})$ using De Morgan's law:

   $P(\overline{A} \cap \overline{B}) = P(\overline{A \cup B}) = 1 - P(A \cup B) = 1 - 0.75 = 0.25$

4. Calculate $P(\overline{B})$:

   $P(\overline{B}) = 1 - P(B) = 1 - 0.6 = 0.4$

5. Calculate $P(\frac{\overline{A}}{\overline{B}})$:

   $P(\frac{\overline{A}}{\overline{B}}) = \frac{P(\overline{A} \cap \overline{B})}{P(\overline{B})} = \frac{0.25}{0.4} = \frac{25}{40} = \frac{5}{8}$

Therefore, $P(\frac{\overline{A}}{\overline{B}}) = \frac{5}{8}$.