Question

A die is thrown once. If E represents the event ‘the number obtained on the die is a multiple of 3’ and F represents the event ‘the number obtained on the die is even’, then tell whether the events E and F are independent.

Hint:

To check if two events are independent, verify if \( P(E \cap F) = P(E) \cdot P(F) \). If it holds, the events are independent.

Answer 1

Step 1: We calculate \( P(E) \), \( P(F) \), and \( P(E \cap F) \): - \( E = \{3, 6\} \), so \( P(E) = \frac{2}{6} = \frac{1}{3} \) - \( F = \{2, 4, 6\} \), so \( P(F) = \frac{3}{6} = \frac{1}{2} \) - \( E \cap F = \{6\} \), so \( P(E \cap F) = \frac{1}{6} \)

Step 2: Check if the events are independent: For independent events, \( P(E \cap F) = P(E) \cdot P(F) \). \[ P(E) \cdot P(F) = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6} \] Since \( P(E \cap F) = \frac{1}{6} \), the events \( E \) and \( F \) are independent.