A bag contains 12 balls of which 3 are red and 5 balls are drawn at random. Find the probability that 3 balls are red.

Option d: 1/22. Solution: Probability of Drawing 3 Red Balls Out of 5 Total balls in the bag = 12 - Red balls = 3 - Non-red balls = 9 Five balls are drawn at random. We want to find the probability that exactly 3 of the drawn balls are red. Step 1: Calculate total ways to draw 5 balls Total ways = (12 choose 5) = 12 × 11 × 10 × 9 × 8/5 × 4 × 3 × 2 × 1 = 792 Step 2: Calculate favorable ways To get exactly 3 red balls, we must choose all 3 red balls and 2 non-red balls: Favorable ways = (3 choose 3) × (9 choose 2) = 1 × 9 × 8/2 = 36 Step 3: Probability calculation P = 36/792 = 1/22 Hence, Option D is the correct answer.

ProbabilityMCQMTP Apr 21Question 3381 of 187

All Questions

A bag contains $\displaystyle 12$ balls of which $\displaystyle 3$ are red and $\displaystyle 5$ balls are drawn at random. Find the probability that $\displaystyle 3$ balls are red.

Options

\displaystyle \frac{3}{132}

\displaystyle \frac{5}{396}

\displaystyle \frac{1}{36}

\displaystyle \frac{1}{22}

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Correct Answer

✅ Option d — $\displaystyle \frac{1}{22}$

All Options:

A $\displaystyle \frac{3}{132}$
B $\displaystyle \frac{5}{396}$
C $\displaystyle \frac{1}{36}$
D $\displaystyle \frac{1}{22}$

Detailed Solution & Explanation

**Probability of Drawing 3 Red Balls Out of 5** Total balls in the bag = 12 - Red balls = 3 - Non-red balls = 9 Five balls are drawn at random. We want to find the probability that exactly 3 of the drawn balls are red. **Step 1: Calculate total ways to draw 5 balls**

\text{Total ways} = \binom{12}{5} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1} = 792

**Step 2: Calculate favorable ways** To get exactly 3 red balls, we must choose all 3 red balls and 2 non-red balls:

\text{Favorable ways} = \binom{3}{3} \times \binom{9}{2} = 1 \times \frac{9 \times 8}{2} = 36

**Step 3: Probability calculation**

P = \frac{36}{792} = \frac{1}{22}

Hence, **Option D** 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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A bag contains 12\displaystyle 1212 balls of which 3\displaystyle 33 are red and 5\displaystyle 55 balls are drawn at random. Find the probability that 3\displaystyle 33 balls are red.