ProbabilityMCQMTP Dec 2023 Series IIQuestion 2832 of 295
All Questions

The probability that A speaks truth is 45\displaystyle \frac{4}{5} while this probability for B is 34\displaystyle \frac{3}{4}. The probability that they contradict each other when asked to speak on a fact is

Options

A320\displaystyle \frac{3}{20}
B15\displaystyle \frac{1}{5}
C720\displaystyle \frac{7}{20}
D35\displaystyle \frac{3}{5}
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Correct Answer

✅ Option c — 720\displaystyle \frac{7}{20}

All Options:

  • A320\displaystyle \frac{3}{20}
  • B15\displaystyle \frac{1}{5}
  • C720\displaystyle \frac{7}{20}
  • D35\displaystyle \frac{3}{5}

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Detailed Solution & Explanation

Let TA\displaystyle T_A be the event that A speaks the truth, and TB\displaystyle T_B be the event that B speaks the truth. We are given: - P(TA)=45  ⟹  P(TA′)=1−45=15\displaystyle P(T_A) = \frac{4}{5} \implies P(T_A') = 1 - \frac{4}{5} = \frac{1}{5} - P(TB)=34  ⟹  P(TB′)=1−34=14\displaystyle P(T_B) = \frac{3}{4} \implies P(T_B') = 1 - \frac{3}{4} = \frac{1}{4} A and B contradict each other when one of them speaks the truth and the other tells a lie. Since their truth-telling behaviors are independent events, we have: P(contradiction)=P(TA∩TB′)+P(TA′∩TB)=P(TA)P(TB′)+P(TA′)P(TB)\displaystyle P(\text{contradiction}) = P(T_A \cap T_B') + P(T_A' \cap T_B) = P(T_A)P(T_B') + P(T_A')P(T_B). Substitute the probabilities: P(contradiction)=(45×14)+(15×34)=420+320=720\displaystyle P(\text{contradiction}) = \left(\frac{4}{5} \times \frac{1}{4}\right) + \left(\frac{1}{5} \times \frac{3}{4}\right) = \frac{4}{20} + \frac{3}{20} = \frac{7}{20}. Hence, **Option C** 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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