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Question
A pair of dice is thrown. Find the probability of getting 7 as the sum if it is known that the second die always exhibits a prime number.
Solution
Consider the given events
A = A prime number appears on second die.
B = The sum of the numbers on two dice is 7.
Clearly,
A = {(1, 2), (1, 3), (1, 5), (2, 2), (2, 3)(2, 5), (3, 2), (3, 3), (3, 5) (4, 2), (4, 3), (4, 5),(5, 2), (5, 3), (5, 5), (6, 2), (6, 3),(6, 5)}
B = {(2, 5), (5, 2), (3, 4), (4, 3), (1, 6), (6, 1)}
\[\text{ Now } , \]
\[A \cap B = {(2, 5), (5, 2),(4, 3)}\]
\[\therefore \text{ Required probability } = P\left( B/A \right) = \frac{n\left( A \cap B \right)}{n\left( A \right)} = \frac{3}{18} = \frac{1}{6}\]
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