The probability that a student is not a swimmer is $1/5$ then the probability that out of five students four are swimmer is

The correct answer is Option c: $^5C_4 \left(\frac{4}{5}\right)^4 \left(\frac{1}{5}\right)^1$. **Probability of Exactly 4 Swimmers out of 5 Students** **Given:** - $P(\text{not a swimmer}) = \\frac{1}{5}$, so $q = \\frac{1}{5}$ - $P(\text{swimmer}) = 1 - \\frac{1}{5} = \\frac{4}{5}$, so $p = \\frac{4}{5}$ - $n = 5$ students, we want exactly $r = 4$ swimmers **Using Binomial Probability Formula:** $$P(X = r) = \binom{n}{r} p^r q^{n-r}$$ $$P(X = 4) = \binom{5}{4} \left(\\frac{4}{5}\right)^4 \left(\\frac{1}{5}\right)^{5-4}$$ $$= {}^5C_4 \left(\\frac{4}{5}\right)^4 \left(\\frac{1}{5}\right)^1$$ **Checking the options:** - **Option A:** ${}^5C_1 \left(\\frac{1}{5}\right)^1 \left(\\frac{4}{5}\right)^4$ — This uses ${}^5C_1$ instead of ${}^5C_4$. Note ${}^5C_1 = {}^5C_4 = 5$, so numerically equal, but the expression is set up conceptually wrong (selecting 1 non-swimmer vs. 4 swimmers). The standard form for selecting 4 swimmers is ${}^5C_4$. - **Option B:** Wrong powers assigned — $p$ and $q$ swapped incorrectly. - **Option C:** ${}^5C_4 \left(\\frac{4}{5}\right)^4 \left(\\frac{1}{5}\right)^1$ — This is **exactly** the binomial formula: choosing 4 swimmers from 5, each swimmer has probability $\\frac{4}{5}$, and the remaining 1 student is not a swimmer with probability $\\frac{1}{5}$. ✓ Hence, **Option C** is the correct answer.

Theoretical DistributionsMCQPYQ Nov 18Question 3417 of 230

All Questions

The probability that a student is not a swimmer is $\displaystyle 1/5$ then the probability that out of five students four are swimmer is

Options

\displaystyle ^5C_1 \left(\frac{1}{5}\right)^1 \left(\frac{4}{5}\right)^4

\displaystyle ^5C_1 \left(\frac{4}{5}\right)^1 \left(\frac{1}{5}\right)^4

\displaystyle ^5C_4 \left(\frac{4}{5}\right)^4 \left(\frac{1}{5}\right)^1

DNone of the above

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Correct Answer

✅ Option c — $\displaystyle ^5C_4 \left(\frac{4}{5}\right)^4 \left(\frac{1}{5}\right)^1$

All Options:

A $\displaystyle ^5C_1 \left(\frac{1}{5}\right)^1 \left(\frac{4}{5}\right)^4$
B $\displaystyle ^5C_1 \left(\frac{4}{5}\right)^1 \left(\frac{1}{5}\right)^4$
C $\displaystyle ^5C_4 \left(\frac{4}{5}\right)^4 \left(\frac{1}{5}\right)^1$
DNone of the above

Detailed Solution & Explanation

**Probability of Exactly 4 Swimmers out of 5 Students** **Given:** -

\displaystyle P(\text{not a swimmer}) = \\frac{1}{5}

, so

\displaystyle q = \\frac{1}{5}

\displaystyle P(\text{swimmer}) = 1 - \\frac{1}{5} = \\frac{4}{5}

, so

\displaystyle p = \\frac{4}{5}

\displaystyle n = 5

students, we want exactly

\displaystyle r = 4

swimmers **Using Binomial Probability Formula:**

P(X = r) = \binom{n}{r} p^r q^{n-r}

P(X = 4) = \binom{5}{4} \left(\\frac{4}{5}\right)^4 \left(\\frac{1}{5}\right)^{5-4}

= {}^5C_4 \left(\\frac{4}{5}\right)^4 \left(\\frac{1}{5}\right)^1

**Checking the options:** - **Option A:**

\displaystyle {}^5C_1 \left(\\frac{1}{5}\right)^1 \left(\\frac{4}{5}\right)^4

— This uses

\displaystyle {}^5C_1

instead of

\displaystyle {}^5C_4

. Note

\displaystyle {}^5C_1 = {}^5C_4 = 5

, so numerically equal, but the expression is set up conceptually wrong (selecting 1 non-swimmer vs. 4 swimmers). The standard form for selecting 4 swimmers is

\displaystyle {}^5C_4

. - **Option B:** Wrong powers assigned —

\displaystyle p

and

\displaystyle q

swapped incorrectly. - **Option C:**

\displaystyle {}^5C_4 \left(\\frac{4}{5}\right)^4 \left(\\frac{1}{5}\right)^1

— This is **exactly** the binomial formula: choosing 4 swimmers from 5, each swimmer has probability

\displaystyle \\frac{4}{5}

, and the remaining 1 student is not a swimmer with probability

\displaystyle \\frac{1}{5}

. ✓ Hence, **Option C** 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.

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The probability that a student is not a swimmer is 1/5\displaystyle 1/51/5 then the probability that out of five students four are swimmer is