Question

Two friends, Alka and Alkit appear for an interview for two vacancies for the same post in an Organisation. The probability of Alka's selection is $1/7$ and Alkit is $1/5$. What is the probability that only one of them will be selected ?

• A. $1/35$
• B. $8/15$
• C. $4/5$
• D. $4/35$
• E. $2/7$

Hint:

For probability of "Exactly One" happening out of two independent events $A$ and $B$, the formula is always $P(A)P(B') + P(A')P(B)$.

Answer 1

Answer: (E)

Step 1: Note individual probabilities of selection and rejection.
Probability Alka is selected, $P(A) = 1/7 \Rightarrow$ Rejected $P(A') = 1 - 1/7 = 6/7$.
Probability Alkit is selected, $P(B) = 1/5 \Rightarrow$ Rejected $P(B') = 1 - 1/5 = 4/5$.

Step 2: Define the desired outcome.
"Only one of them is selected" means exactly one of two mutually exclusive events occurs:
Event 1: Alka is selected AND Alkit is rejected $\rightarrow P(A) \times P(B')$
Event 2: Alka is rejected AND Alkit is selected $\rightarrow P(A') \times P(B)$

Step 3: Calculate the combined probability.
$P(\text{only one}) = (\frac{1}{7} \times \frac{4}{5}) + (\frac{6}{7} \times \frac{1}{5}) = \frac{4}{35} + \frac{6}{35} = \frac{10}{35} = \frac{2}{7}$.