For a binomial distribution, the variance is 0.2 and the mean is 0.6 . The probability of getting 3 successes out of a trial of 5 is .

Option a: 80/35. Solution: Given: Variance = 0.2, Mean = 0.6, and the number of trials n = 5. 1: Find q using the relationship between variance and mean. = = 0.2/0.6 = 1/3 2: Find p. = 1 - q = 1 - 1/3 = 2/3 3: Confirm n = 5 (given directly in the problem). 4: Apply the Binomial Probability formula for X = 3.(X = 3) = (5 choose 3) p3 q2 5: Compute the binomial coefficient.(5 choose 3) = 5!/3! × 2! = 10 6: Substitute values.(X = 3) = 10 × (2/3)3 × (1/3)2 7: Simplify.= 10 × 8/27 × 1/9= 10 × 8/243= 80/243, Option A is the correct answer.

Theoretical DistributionsMCQPYQ June 24Question 3431 of 230

All Questions

For a binomial distribution, the variance is $\displaystyle 0.2$ and the mean is $\displaystyle 0.6$ . The probability of getting $\displaystyle 3$ successes out of a trial of $\displaystyle 5$ is _____.

Options

\displaystyle \frac{80}{3^5}

\displaystyle \frac{40}{3^5}

\displaystyle \frac{20}{3^5}

\displaystyle \frac{160}{3^5}

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Correct Answer

✅ Option a — $\displaystyle \frac{80}{3^5}$

All Options:

A $\displaystyle \frac{80}{3^5}$
B $\displaystyle \frac{40}{3^5}$
C $\displaystyle \frac{20}{3^5}$
D $\displaystyle \frac{160}{3^5}$

Detailed Solution & Explanation

\displaystyle Given: Variance = 0.2, Mean = 0.6, and the number of trials n = 5.\n\nStep 1: Find q using the relationship between variance and mean.\nq = \\frac{\\text{Variance}}{\\text{Mean}} = \\frac{0.2}{0.6} = \\frac{1}{3}\n\nStep 2: Find p.\np = 1 - q = 1 - \\frac{1}{3} = \\frac{2}{3}\n\nStep 3: Confirm n = 5 (given directly in the problem).\n\nStep 4: Apply the Binomial Probability formula for X = 3.\nP(X = 3) = \\binom{5}{3} p^3 q^2\n\nStep 5: Compute the binomial coefficient.\n\\binom{5}{3} = \\frac{5!}{3! \\times 2!} = 10\n\nStep 6: Substitute values.\nP(X = 3) = 10 \\times \\left(\\frac{2}{3}\\right)^3 \\times \\left(\\frac{1}{3}\\right)^2\n\nStep 7: Simplify.\n= 10 \\times \\frac{8}{27} \\times \\frac{1}{9}\n= 10 \\times \\frac{8}{243}\n= \\frac{80}{243}\n\nHence, **Option A** is the correct answer.

About This Chapter: Theoretical Distributions

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Binomial, Poisson, Normal Distribution

This chapter covers Binomial, Poisson, Normal Distribution and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 4-6 Marks weightage. Focus on understanding core concepts rather than memorizing.

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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For a binomial distribution, the variance is 0.2\displaystyle 0.20.2 and the mean is 0.6\displaystyle 0.60.6. The probability of getting 3\displaystyle 33 successes out of a trial of 5\displaystyle 55 is _____.