Question

Sonal and Meenal appear in an interview for same post having two vacancies. If \(\frac 17\) is Sonal's probability of selection and \(\frac15\) is Meenal's probability of selection then what is the probability that only one of them is selected?

• A. \(\frac17\)
• B. \(\frac27\)
• C. \(\frac35\)
• D. \(\frac15\)

Answer 1

The Correct Option is B

Given:
\[ P(S) = \frac{1}{7} \]
\[ P(M) = \frac{1}{5} \]
We need to find the probability that only one of them is selected. There are two cases for this:
1. Sonal is selected, and Meenal is not selected.
2. Meenal is selected, and Sonal is not selected.
The probability that Sonal is selected and Meenal is not selected:
\[ P(S \text{ and } \neg M) = P(S) \times (1 - P(M)) \]
\[ = \frac{1}{7} \times \left(1 - \frac{1}{5}\right) \]
\[ = \frac{1}{7} \times \frac{4}{5} \]
\[ = \frac{4}{35} \]
The probability that Meenal is selected and Sonal is not selected:
\[ P(M \text{ and } \neg S) = P(M) \times (1 - P(S)) \]
\[ = \frac{1}{5} \times \left(1 - \frac{1}{7}\right) \]
\[ = \frac{1}{5} \times \frac{6}{7} \]
\[ = \frac{6}{35} \]
The total probability that only one of them is selected is the sum of the two probabilities:
\[ P(\text{only one is selected}) = P(S \text{ and } \neg M) + P(M \text{ and } \neg S) \]
\[ = \frac{4}{35} + \frac{6}{35} \]
\[ = \frac{10}{35} \]
\[ = \frac{2}{7} \]
Thus, the correct answer is:
B. 2/7

Answer 2

Only one of the two candidates is selected means exactly one of the events occurs. This probability can be written directly as:
\[P(\text{exactly one}) = P(S)+P(M)-2\,P(S)P(M)\]

Substituting \(P(S)=\frac{1}{7}\) and \(P(M)=\frac{1}{5}\):
\[P(\text{exactly one}) = \frac{1}{7}+\frac{1}{5}-2\left(\frac{1}{7}\right)\left(\frac{1}{5}\right) = \frac{5}{35}+\frac{7}{35}-\frac{2}{35} = \frac{10}{35} = \frac{2}{7}\]

1. Option A \(\frac{1}{7}\): Does not equal \(\frac{2}{7}\).
2. Option B \(\frac{2}{7}\): Matches the computed value exactly.
3. Option C \(\frac{3}{5}\): Does not equal \(\frac{2}{7}\).
4. Option D \(\frac{1}{5}\): Does not equal \(\frac{2}{7}\).

The probability that only one of them is selected is \(\frac{2}{7}\).

Hence, the correct answer is Option B: \(\frac{2}{7}\).