Question

If \(P(A)=\dfrac{7}{13}\), \(P(B)=\dfrac{9}{13}\) and \(P(A\cap B)=\dfrac{4}{13}\), then find \(P(A\cup B)\) and \(P(A/B)\).

Hint:

Use \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\) and \(P(A/B)=P(A\cap B)/P(B)\).

Answer 1

Step 1: Understanding the Concept:
We are given probabilities of two events A and B and their intersection.
We need the union probability and the conditional probability of A given B.

Step 2: Key Formula or Approach:
Use the addition rule \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\) and the conditional probability formula \(P(A/B)=\dfrac{P(A\cap B)}{P(B)}\).

Step 3: Detailed Explanation:
First find the union probability by substituting the given values:
\[ P(A\cup B) = \frac{7}{13}+\frac{9}{13}-\frac{4}{13} = \frac{7+9-4}{13} = \frac{12}{13} \]
Next find the conditional probability of A given B, which is the intersection divided by P(B):
\[ P(A/B) = \frac{P(A\cap B)}{P(B)} = \frac{4/13}{9/13} = \frac{4}{9} \]

Final Answer:
Both results follow directly from the standard formulas of probability. \[ \boxed{P(A\cup B)=\frac{12}{13}, \quad P(A/B)=\frac{4}{9}} \]