Question

Let Ajay will not appear in JEE exam with probability $p = \frac{2}{7}$, while both Ajay and Vijay will appear in the exam with probability $q = \frac{1}{5}$. Then the probability that Ajay will appear in the exam and Vijay will not appear is:

• A. $\frac{9}{35}$
• B. $\frac{18}{35}$
• C. $\frac{24}{35}$
• D. $\frac{3}{35}$

Answer 1

The Correct Option is B

Given:
\[ P(\text{Ajay does not appear}) = p = \frac{2}{7}, \quad P(\text{Ajay and Vijay both appear}) = q = \frac{1}{5} \]

Let:
\[ P(\text{Ajay appears}) = 1 - p = 1 - \frac{2}{7} = \frac{5}{7} \]
Let \( P(\text{Vijay appears}) = v \). The probability that both Ajay and Vijay appear is given by:
\[ P(\text{Ajay appears}) \times P(\text{Vijay appears}) = q \]
Substituting the given values:
\[ \frac{5}{7} \times v = \frac{1}{5} \]
Solving for \( v \):
\[ v = \frac{1}{5} \times \frac{7}{5} = \frac{7}{25} \]
Thus, the probability that Vijay does not appear is:
\[ P(\text{Vijay does not appear}) = 1 - v = 1 - \frac{7}{25} = \frac{18}{25} \]

Finding the Desired Probability
The probability that Ajay will appear in the exam and Vijay will not appear is given by:
\[ P(\text{Ajay appears}) \times P(\text{Vijay does not appear}) = \frac{5}{7} \times \frac{18}{25} \]
Calculating the product:
\[ P(\text{Ajay appears and Vijay does not appear}) = \frac{5 \times 18}{7 \times 25} = \frac{90}{175} = \frac{18}{35} \]

Conclusion: The probability that Ajay will appear in the exam and Vijay will not appear is \( \frac{18}{35} \).