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Question
Prove that in throwing a pair of dice, the occurrence of the number 4 on the first die is independent of the occurrence of 5 on the second die.
Solution
\[\text{ Total number of events } = 36\]
\[P\left( 4 \text{ on first die } \right) = P\left( A \right) = \frac{6}{36} = \frac{1}{6}\]
\[P\left( 5 \text{ on second die } \right) = P\left( B \right) = \frac{6}{36} = \frac{1}{6}\]
\[P\left( A \cap B \right) = \frac{1}{36}\]
\[P\left( A \cap B \right) = P\left( A \right)P\left( B \right)\]
\[\text{ Thus, A and B are independent events } .\]
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