Question

Let $S = \{2,\,5,\,8,\,11,\,14,\,17,\,20,\,23\}$. Two integers $m,\,n$ are chosen one by one from $S$ with replacement. Then the probability that $mn$ is odd, is

• A. $\dfrac{3}{4}$
• B. $\dfrac{3}{7}$
• C. $\dfrac{3}{5}$
• D. $\dfrac{3}{7}$
• E. $\dfrac{3}{5}$

Hint:

For the product $mn$ to be odd, both $m$ and $n$ must be odd. Count odd elements in the set carefully. Since the selection is with replacement, draws are independent, so multiply the individual probabilities.

Answer 1

The Correct Option is B

Step 1:
$S = \{2,5,8,11,14,17,20,23\}$, number of odd elements $= 4$, total elements $= 8$.


Step 2:
$P(m \text{ odd}) = \dfrac{4}{8} = \dfrac{1}{2}$.


Step 3:
$P(n \text{ odd} \mid m \text{ odd}) = \dfrac{3}{7}$.


Step 4:
$P(mn \text{ odd}) = \dfrac{4}{8} \times \dfrac{3}{7} = \dfrac{3}{7}$.
Final Answer:
Option B: $\dfrac{3}{7}$.