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Question
A die is thrown twice and the sum of the numbers appearing is observed to be 8. What is the conditional probability that the number 5 has appeared at least once?
Solution
Consider the given events.
A = 5 appears on the die at least once
B = The sum of the numbers on two dice is 8.
Clearly,
A = {(1, 5),(2, 5),(3, 5),(4, 5),(5, 5), (6, 5), (5, 1), (5, 2), (5, 3), (5, 4), (5, 6)}
B = {(2, 6), (3, 5), (4, 4), (5, 3),(6, 2)}
\[\text{ Now } , \]
\[A \cap B = \left\{ \left( 3, 5 \right), \left( 5, 3 \right) \right\}\]
\[ \therefore \text{ Required probability } = P\left( A/B \right) = \frac{n\left( A \cap B \right)}{n\left( B \right)} = \frac{2}{5}\]
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