ProbabilityMCQMTP Nov 18Question 2818 of 295
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A probability in statistics is given to five students A, B, C, D and E. Their chances of is 12\displaystyle \frac{1}{2}, 13\displaystyle \frac{1}{3}, 14\displaystyle \frac{1}{4}, 15\displaystyle \frac{1}{5}, 16\displaystyle \frac{1}{6}. What's the probability that the problem will be solved.

Options

A16\displaystyle \frac{1}{6}
B56\displaystyle \frac{5}{6}
C1
DNone
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Correct Answer

✅ Option b — 56\displaystyle \frac{5}{6}

All Options:

  • A16\displaystyle \frac{1}{6}
  • B56\displaystyle \frac{5}{6}
  • C1
  • DNone

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Detailed Solution & Explanation

Let A\displaystyle A, B\displaystyle B, C\displaystyle C, D\displaystyle D, and E\displaystyle E represent the independent events that students A, B, C, D, and E solve the problem, respectively. 1. The individual probabilities of solving the problem are: - P(A)=12\displaystyle P(A) = \frac{1}{2} - P(B)=13\displaystyle P(B) = \frac{1}{3} - P(C)=14\displaystyle P(C) = \frac{1}{4} - P(D)=15\displaystyle P(D) = \frac{1}{5} - P(E)=16\displaystyle P(E) = \frac{1}{6} 2. The probabilities that each student fails to solve the problem are: - P(A′)=1−12=12\displaystyle P(A') = 1 - \frac{1}{2} = \frac{1}{2} - P(B′)=1−13=23\displaystyle P(B') = 1 - \frac{1}{3} = \frac{2}{3} - P(C′)=1−14=34\displaystyle P(C') = 1 - \frac{1}{4} = \frac{3}{4} - P(D′)=1−15=45\displaystyle P(D') = 1 - \frac{1}{5} = \frac{4}{5} - P(E′)=1−16=56\displaystyle P(E') = 1 - \frac{1}{6} = \frac{5}{6} 3. Since the events are independent, the probability that none of the students solve the problem is the product of their individual failure probabilities: P(None solves)=P(A′)×P(B′)×P(C′)×P(D′)×P(E′)P(\text{None solves}) = P(A') \times P(B') \times P(C') \times P(D') \times P(E') P(None solves)=12×23×34×45×56=16P(\text{None solves}) = \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \frac{4}{5} \times \frac{5}{6} = \frac{1}{6} 4. The probability that the problem is solved is the complement of the probability that none of them solve it: P(Solved)=1−P(None solves)=1−16=56P(\text{Solved}) = 1 - P(\text{None solves}) = 1 - \frac{1}{6} = \frac{5}{6} 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.

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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