ProbabilityMCQMTP June 24 Series IIQuestion 2846 of 295

A dice is rolled thrice, if getting a four is considered a success, find the variance of the probability distribution of number of successes

Options

\displaystyle \frac{1}{2}

\displaystyle \frac{5}{24}

\displaystyle \frac{5}{12}

\displaystyle \frac{7}{12}

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Correct Answer

✅ Option c — $\displaystyle \frac{5}{12}$

All Options:

A $\displaystyle \frac{1}{2}$
B $\displaystyle \frac{5}{24}$
C $\displaystyle \frac{5}{12}$
D $\displaystyle \frac{7}{12}$

Detailed Solution & Explanation

This scenario follows a Binomial distribution since there are a fixed number of independent trials (

\displaystyle n = 3

), and each trial has only two outcomes: success (rolling a four) or failure (rolling any other number). - Number of trials,

\displaystyle n = 3

- Probability of success in a single trial (rolling a 4 on a six-sided die),

\displaystyle p = \frac{1}{6}

- Probability of failure in a single trial,

\displaystyle q = 1 - p = 1 - \frac{1}{6} = \frac{5}{6}

For a Binomial distribution, the variance is given by:

\displaystyle \text{Variance} = n p q

\displaystyle \text{Variance} = 3 \times \frac{1}{6} \times \frac{5}{6} = \frac{15}{36} = \frac{5}{12}

. 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.

Variance

A measure of dispersion equal to the average of the squared deviations of each data point from the mean. Formula: σ² = Σ(xᵢ − x̄)² / n. Standard Deviation = √Variance.

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