ProbabilityMCQMTP Jun 24 Series IIQuestion 2793 of 295

If two letters are taken at random from the word HOME, what is the Probability that none of the letters would be vowels?

Options

\displaystyle \frac{1}{6}

\displaystyle \frac{1}{2}

\displaystyle \frac{1}{3}

\displaystyle \frac{1}{4}

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Correct Answer

✅ Option a — $\displaystyle \frac{1}{6}$

All Options:

A $\displaystyle \frac{1}{6}$
B $\displaystyle \frac{1}{2}$
C $\displaystyle \frac{1}{3}$
D $\displaystyle \frac{1}{4}$

Detailed Solution & Explanation

We are choosing 2 letters at random from the 4-letter word 'HOME'. 1. The word HOME contains 4 distinct letters: {H, O, M, E}. - Vowels = {O, E} (2 vowels) - Consonants = {H, M} (2 consonants) 2. The total number of ways to choose 2 letters from 4 is:

\binom{4}{2} = 6

3. The number of ways to choose 2 letters such that none of them are vowels (i.e., both must be consonants chosen from {H, M}) is:

\binom{2}{2} = 1

4. The probability is:

P(\text{no vowels}) = \frac{1}{6}

This corresponds to Option A. 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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