ProbabilityMCQMTP Nov 19Question 2815 of 295

The probability that a person travels by a plane is $\displaystyle \frac{1}{5}$ and that he travels by train is $\displaystyle \frac{2}{3}$ . Find the probability of his traveling neither by plane nor by train?

Options

\displaystyle \frac{13}{15}

\displaystyle \frac{2}{15}

\displaystyle \frac{1}{15}

DNone of these

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Correct Answer

✅ Option b — $\displaystyle \frac{2}{15}$

All Options:

A $\displaystyle \frac{13}{15}$
B $\displaystyle \frac{2}{15}$
C $\displaystyle \frac{1}{15}$
DNone of these

Detailed Solution & Explanation

Let

\displaystyle P

represent the event that the person travels by plane, and

\displaystyle T

represent the event that the person travels by train. 1. The given probabilities are: -

\displaystyle P(P) = \frac{1}{5}

\displaystyle P(T) = \frac{2}{3}

2. Assuming traveling by plane and traveling by train are mutually exclusive events for the journey, the probability that he travels by plane or train is:

P(P \cup T) = P(P) + P(T) = \frac{1}{5} + \frac{2}{3} = \frac{3 + 10}{15} = \frac{13}{15}

3. The probability that he travels by neither plane nor train is the complement of

\displaystyle P(P \cup T)

P(\text{neither}) = 1 - P(P \cup T) = 1 - \frac{13}{15} = \frac{2}{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.

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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The probability that a person travels by a plane is 15\displaystyle \frac{1}{5}51​ and that he travels by train is 23\displaystyle \frac{2}{3}32​. Find the probability of his traveling neither by plane nor by train?