ProbabilityPYQ May 25Question 4084 of 187
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A problem is given to 5 students P, Q, R, S and T. If the probability of solving the problem individually is 12\displaystyle \frac{1}{2}, 13\displaystyle \frac{1}{3}, 23\displaystyle \frac{2}{3}, 15\displaystyle \frac{1}{5} and 16\displaystyle \frac{1}{6} respectively, then find the probability that the problem is solved.

Options

A0.47
B0.93
C0.57
D0.27
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Correct Answer

✅ Option b — 0.93

All Options:

  • A0.47
  • B0.93
  • C0.57
  • D0.27

Detailed Solution & Explanation

Let the events that students P,Q,R,S,T\displaystyle P, Q, R, S, T solve the problem be A,B,C,D,E\displaystyle A, B, C, D, E respectively. The individual probabilities are:
P(A)=12,P(B)=13,P(C)=23,P(D)=15,P(E)=16P(A) = \frac{1}{2}, \quad P(B) = \frac{1}{3}, \quad P(C) = \frac{2}{3}, \quad P(D) = \frac{1}{5}, \quad P(E) = \frac{1}{6}
The problem is solved if at least one student solves it. To find this probability, it is easier to compute the probability of the complementary event (that none of the students solve the problem):
The probabilities of the students failing to solve the problem are:
P(A′)=1−12=12P(A') = 1 - \frac{1}{2} = \frac{1}{2} P(B′)=1−13=23P(B') = 1 - \frac{1}{3} = \frac{2}{3} P(C′)=1−23=13P(C') = 1 - \frac{2}{3} = \frac{1}{3} P(D′)=1−15=45P(D') = 1 - \frac{1}{5} = \frac{4}{5} P(E′)=1−16=56P(E') = 1 - \frac{1}{6} = \frac{5}{6}
Since the students work individually, these events are independent. The probability that none of them solve the problem is:
P(None)=P(A′)×P(B′)×P(C′)×P(D′)×P(E′)P(\text{None}) = P(A') \times P(B') \times P(C') \times P(D') \times P(E') P(None)=12×23×13×45×56P(\text{None}) = \frac{1}{2} \times \frac{2}{3} \times \frac{1}{3} \times \frac{4}{5} \times \frac{5}{6} P(None)=1×2×1×4×52×3×3×5×6=40540=454=227≈0.074P(\text{None}) = \frac{1 \times 2 \times 1 \times 4 \times 5}{2 \times 3 \times 3 \times 5 \times 6} = \frac{40}{540} = \frac{4}{54} = \frac{2}{27} \approx 0.074
The probability that the problem is solved is:
P(Solved)=1−P(None)=1−227=2527≈0.9259≈0.93P(\text{Solved}) = 1 - P(\text{None}) = 1 - \frac{2}{27} = \frac{25}{27} \approx 0.9259 \approx 0.93
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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