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:
- $\frac{9}{35}$
- $\frac{18}{35}$
- $\frac{24}{35}$
- $\frac{3}{35}$
The Correct Option is B
Approach Solution - 1
To solve the problem, let's define the different probabilities given and calculate the required probability step-by-step:
- 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 \(1 - p = 1 - \frac{2}{7} = \frac{5}{7}\).
- The probability that both Ajay and Vijay will appear in the exam is given by \(q = \frac{1}{5}\).
- We need to find the probability that Ajay will appear in the exam and Vijay will not appear. We can denote this as \(P(A \cap V')\).
- Using the total probability for Ajay's appearance:
- \(P(A \cap V) + P(A \cap V') = P(A)\)
- Substituting the known values: \(\frac{1}{5} + P(A \cap V') = \frac{5}{7}\)
- 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 \(\frac{18}{35}\).
Approach Solution -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 \( \frac{18}{35} \).
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