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

To solve the problem, let's define the different probabilities given and calculate the required probability step-by-step:

1. The probability that Ajay will not appear in the JEE exam is given by \(p = \frac{2}{7}\). Thus, the probability that Ajay will appear in the exam is \(1 - p = 1 - \frac{2}{7} = \frac{5}{7}\).
2. The probability that both Ajay and Vijay will appear in the exam is given by \(q = \frac{1}{5}\).
3. We need to find the probability that Ajay will appear in the exam and Vijay will not appear. We can denote this as \(P(A \cap V')\).
4. Using the total probability for Ajay's appearance:
   • \(P(A \cap V) + P(A \cap V') = P(A)\)
   • Substituting the known values: \(\frac{1}{5} + P(A \cap V') = \frac{5}{7}\)
5. Solve the equation for \(P(A \cap V')\):
   • \(P(A \cap V') = \frac{5}{7} - \frac{1}{5}\)
   • To subtract these fractions, find a common denominator (35):
   • \(\frac{5}{7} = \frac{25}{35} \quad \text{and} \quad \frac{1}{5} = \frac{7}{35}\)
   • Therefore, \(P(A \cap V') = \frac{25}{35} - \frac{7}{35} = \frac{18}{35}\)

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

The correct answer is \(\frac{18}{35}\).

Answer 2

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 \( \frac{18}{35} \).