Question

A die is thrown once. The number on the die is a multiple of 3 is denoted by \( E \), and the number on the die is even is denoted by \( F \). Are \( E \) and \( F \) independent events?

Hint:

Two events \( A \) and \( B \) are independent if \( P(A \cap B) = P(A) P(B) \).

Answer 1

Step 1: Find probabilities. Sample space: \( S = \{1,2,3,4,5,6\} \). \[ P(E) = \frac{\text{multiples of 3}}{\text{total outcomes}} = \frac{2}{6} = \frac{1}{3} \] \[ P(F) = \frac{\text{even numbers}}{\text{total outcomes}} = \frac{3}{6} = \frac{1}{2} \] \[ P(E \cap F) = \frac{\text{even multiples of 3}}{\text{total outcomes}} = \frac{1}{6} \] Step 2: Check independence condition. \[ P(E) \cdot P(F) = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6} \] Since \( P(E \cap F) = P(E) P(F) \), events are independent.