Question

Two persons appear for an interview for two posts. The probability of the person A's selection is 1/5 and that of B is 1/7. The probability of atleast one of them is selected

• A. \(\frac{9}{5}\)
• B. \(\frac{12}{35}\)
• C. \(\frac{2}{7}\)
• D. \(\frac{11}{35}\)

Hint:

Addition formula: $P(A \cup B) = \frac{1}{5} + \frac{1}{7} - \left(\frac{1}{5} \times \frac{1}{7}\right) = \frac{7 + 5 - 1}{35} = \frac{11}{35}$.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The probability of the union of two independent events is calculated using the complement rule or the addition theorem of probability.
Key Formula or Approach:
\[ P(A \cup B) = 1 - P(A' \cap B') = 1 - [1 - P(A)][1 - P(B)] \]
or \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Step 2: Detailed Explanation:
Given:
- Probability of A's selection: \(P(A) = \frac{1}{5} \implies P(A') = 1 - \frac{1}{5} = \frac{4}{5}\)
- Probability of B's selection: \(P(B) = \frac{1}{7} \implies P(B') = 1 - \frac{1}{7} = \frac{6}{7}\)
Since selections are independent events:
Probability that neither is selected:
\[ P(A' \cap B') = P(A') \times P(B') = \frac{4}{5} \times \frac{6}{7} = \frac{24}{35} \]
Probability that at least one is selected:
\[ P(\text{At least one}) = 1 - P(A' \cap B') = 1 - \frac{24}{35} = \frac{11}{35} \]

Step 3: Final Answer:
Therefore, the probability of at least one being selected is \(\frac{11}{35}\), corresponding to option (D).