ProbabilityMCQPYQ June 24Question 3341 of 187
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A question in statistics is given to three students A, B and C. Their chances of solving the question are 1/3,1/5\displaystyle 1/3, 1/5 and 1/7\displaystyle 1/7 respectively. The probability that the question would be solved is

Options

A19/35\displaystyle 19/35
B16/35\displaystyle 16/35
C1/105\displaystyle 1/105
D104/105\displaystyle 104/105
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Correct Answer

✅ Option a — 19/35\displaystyle 19/35

All Options:

  • A19/35\displaystyle 19/35
  • B16/35\displaystyle 16/35
  • C1/105\displaystyle 1/105
  • D104/105\displaystyle 104/105

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Detailed Solution & Explanation

**Probability of the Question Being Solved** Let A,B,\displaystyle A, B, and C\displaystyle C be the events that students A, B, and C solve the question, respectively. We are given: - P(A)=13⇒P(Aˉ)=1−13=23\displaystyle P(A) = \frac{1}{3} \Rightarrow P(\bar{A}) = 1 - \frac{1}{3} = \frac{2}{3} - P(B)=15⇒P(Bˉ)=1−15=45\displaystyle P(B) = \frac{1}{5} \Rightarrow P(\bar{B}) = 1 - \frac{1}{5} = \frac{4}{5} - P(C)=17⇒P(Cˉ)=1−17=67\displaystyle P(C) = \frac{1}{7} \Rightarrow P(\bar{C}) = 1 - \frac{1}{7} = \frac{6}{7} The question is solved if at least one student solves it. Using the complement rule: P(solved)=1−P(none solves)P(\text{solved}) = 1 - P(\text{none solves}) Assuming the events are independent, the probability that none of them solves the question is: P(none)=P(Aˉ)×P(Bˉ)×P(Cˉ)P(\text{none}) = P(\bar{A}) \times P(\bar{B}) \times P(\bar{C}) P(none)=23×45×67=48105=1635P(\text{none}) = \frac{2}{3} \times \frac{4}{5} \times \frac{6}{7} = \frac{48}{105} = \frac{16}{35} Now, compute the probability that the question is solved: P(solved)=1−1635=1935P(\text{solved}) = 1 - \frac{16}{35} = \frac{19}{35} This matches Option A. (Note: The official answer key incorrectly lists Option D as correct, which results from a common error of calculating P(none)=P(A)×P(B)×P(C)=1105\displaystyle P(\text{none}) = P(A) \times P(B) \times P(C) = \frac{1}{105} and subtracting it from 1\displaystyle 1). Hence, **Option A** 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.

Key Concepts to Understand

Probability

A numerical measure of the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). Formula: P(A) = Number of favorable outcomes / Total number of equally likely outcomes.

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