ProbabilityPYQ Sept 25Question 4180 of 187
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Two-person X and Y appear in an interview for two vacancies for the same post. The probability of X's selection is 1/5 and that of Y's selection is 1/3. The probability that none of them will be selected is

Options

A7/15
B8/15
C9/15
D10/15
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Correct Answer

✅ Option b — 8/15

All Options:

  • A7/15
  • B8/15
  • C9/15
  • D10/15

Detailed Solution & Explanation

Let A\displaystyle A be the event that X\displaystyle X is selected, and B\displaystyle B be the event that Y\displaystyle Y is selected. We are given their individual selection probabilities:
P(A)=15andP(B)=13P(A) = \frac{1}{5} \quad \text{and} \quad P(B) = \frac{1}{3}
1. **Find the probability of non-selection**:
- Probability that X\displaystyle X is not selected (A′\displaystyle A'):
P(A′)=1−P(A)=1−15=45P(A') = 1 - P(A) = 1 - \frac{1}{5} = \frac{4}{5}
- Probability that Y\displaystyle Y is not selected (B′\displaystyle B'):
P(B′)=1−P(B)=1−13=23P(B') = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3}
2. **Find the probability that none of them are selected**:
Since the selections of X\displaystyle X and Y\displaystyle Y are independent events, their non-selections are also independent. Therefore:
P(None selected)=P(A′∩B′)=P(A′)×P(B′)P(\text{None selected}) = P(A' \cap B') = P(A') \times P(B')
P(None selected)=45×23=815P(\text{None selected}) = \frac{4}{5} \times \frac{2}{3} = \frac{8}{15}
Hence, **Option B** is the correct answer.

About This Chapter: Probability

Paper

Paper 3: Quantitative Aptitude

Weightage

5-7 Marks

Key Topics

Probability Operations, Expected Value

A logic-heavy chapter dealing with random experiments, events (mutually exclusive, exhaustive), set theory probability, conditional probability, and Bayes' Theorem. It forms the basis for Theoretical Distributions.

View Official ICAI Syllabus

Exam Strategy Tip

Always draw a quick Venn Diagram or tree when faced with 'At least one' or 'Only A but not B' wording. It saves you from double-counting.

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