Question

A die is thrown twice. Let us represent the event 'obtaining an odd number on the first throw' by A and the event 'obtaining an odd number on the second throw' by B. Test the independency of the events A and B.

Hint:

To test the independence of two events, check if \( P(A \cap B) = P(A) \cdot P(B) \).

Answer 1

Step 1: The events \( A \) and \( B \) are independent if: \[ P(A \cap B) = P(A) \cdot P(B). \]

Step 2: For a fair die, the probability of rolling an odd number (1, 3, or 5) is \( \frac{3}{6} = \frac{1}{2} \), so: \[ P(A) = P(B) = \frac{1}{2}. \] 

Step 3: The probability of both events occurring (i.e., rolling an odd number on both throws) is: \[ P(A \cap B) = P(\text{odd on first throw and odd on second throw}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}. \] 

Step 4: Now check if \( P(A \cap B) = P(A) \cdot P(B) \): \[ P(A \cap B) = \frac{1}{4}, \quad P(A) \cdot P(B) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}. \] Since they are equal, the events \( A \) and \( B \) are independent.