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:
To address the problem, we will first define the given probabilities and then calculate the required probability through a series of steps:
The probability of Ajay's absence from the JEE exam is given as \(p = \frac{2}{7}\). Consequently, the probability of Ajay's attendance is \(1 - p = 1 - \frac{2}{7} = \frac{5}{7}\).
The probability that both Ajay and Vijay attend the exam is given as \(q = \frac{1}{5}\).
Our objective is to determine the probability that Ajay attends the exam while Vijay does not. This can be represented as \(P(A \cap V')\).
Using the principle of total probability for Ajay's attendance:
We have the equation \(P(A \cap V) + P(A \cap V') = P(A)\).
Substituting the known values into the equation yields: \(\frac{1}{5} + P(A \cap V') = \frac{5}{7}\).
We now solve the equation for \(P(A \cap V')\):
Rearranging the equation, we get \(P(A \cap V') = \frac{5}{7} - \frac{1}{5}\).
To perform the subtraction of these fractions, we find a common denominator, which is 35:
The fractions are converted as follows: \(\frac{5}{7} = \frac{25}{35}\) and \(\frac{1}{5} = \frac{7}{35}\).
Therefore, the subtraction results in \(P(A \cap V') = \frac{25}{35} - \frac{7}{35} = \frac{18}{35}\).
The probability that Ajay attends the exam and Vijay does not is \(\frac{18}{35}\).
