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.

Answer 1

\[\text{ Suppose O represents the event of getting two odd numbers and S represents the event of getting their sum as an even number } . \]

\[\text{ Now } , \]

\[P\left( O/S \right) = \frac{P\left( O \cap S \right)}{P\left( S \right)} = \frac{\frac{{}^5 C_2}{{}^9 C_2}}{\frac{\left( {}^4 C_2 +^5 C_2 \right)}{{}^9 C_2}} = \frac{{}^5 C_2}{{}^4 C_2 +^5 C_2} = \frac{10}{16} = \frac{5}{8}\]

\[\]