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Question
Two numbers are selected at random from integers 1 through 9. If the sum is even, find the probability that both the numbers are odd.
Solution
\[\text{ Suppose O represents the event of getting two odd numbers and S represents the event of getting their sum as an even number } . \]
\[\text{ Now } , \]
\[P\left( O/S \right) = \frac{P\left( O \cap S \right)}{P\left( S \right)} = \frac{\frac{{}^5 C_2}{{}^9 C_2}}{\frac{\left( {}^4 C_2 +^5 C_2 \right)}{{}^9 C_2}} = \frac{{}^5 C_2}{{}^4 C_2 +^5 C_2} = \frac{10}{16} = \frac{5}{8}\]
\[\]
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