ProbabilityMCQMTP Sep 24 Series IIQuestion 3312 of 187

A box contains $\displaystyle 5$ white and $\displaystyle 7$ black balls. Two successive drawn of $\displaystyle 3$ balls are made (i) with replacement (ii) without replacement. The probability that the first draw would produce white balls and the second draw would produce black balls are respectively.

Options

\displaystyle 6/321

and

\displaystyle 3/926

\displaystyle 1/10

and

\displaystyle 1/30

\displaystyle 35/144

and

\displaystyle 35/108

\displaystyle 7/968

and

\displaystyle 5/264

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Correct Answer

✅ Option d — $\displaystyle 7/968$ and $\displaystyle 5/264$

All Options:

A $\displaystyle 6/321$ and $\displaystyle 3/926$
B $\displaystyle 1/10$ and $\displaystyle 1/30$
C $\displaystyle 35/144$ and $\displaystyle 35/108$
D $\displaystyle 7/968$ and $\displaystyle 5/264$

Detailed Solution & Explanation

**Two Successive Draws: 3 White Then 3 Black** Box: 5 white + 7 black = 12 balls total. **(i) With Replacement:**

P(\text{3 white in 1st draw}) = \frac{\binom{5}{3}}{\binom{12}{3}} = \frac{10}{220} = \frac{1}{22}

After replacement, box is same: 5W, 7B, 12 total.

P(\text{3 black in 2nd draw}) = \frac{\binom{7}{3}}{\binom{12}{3}} = \frac{35}{220} = \frac{7}{44}

P_1 = \frac{1}{22} \times \frac{7}{44} = \frac{7}{968}

**(ii) Without Replacement:** After drawing 3 white balls: box has 2 white + 7 black = 9 balls.

P(\text{3 white in 1st draw}) = \frac{\binom{5}{3}}{\binom{12}{3}} = \frac{10}{220} = \frac{1}{22}

P(\text{3 black in 2nd draw | 3 white drawn}) = \frac{\binom{7}{3}}{\binom{9}{3}} = \frac{35}{84} = \frac{5}{12}

P_2 = \frac{1}{22} \times \frac{5}{12} = \frac{5}{264}

So the answers are

\displaystyle \frac{7}{968}

and

\displaystyle \frac{5}{264}

— which is **Option D**. 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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