ProbabilityMCQMTP June 24 Series IIQuestion 3392 of 187

A bag contains 8 red and 5 white balls. Two successive draws of 3 balls are made without replacement. The prob. that the first draw will produce 3 white ball and second 3 red balls is :

Options

\displaystyle \frac{6}{255}

\displaystyle \frac{5}{548}

\displaystyle \frac{7}{429}

\displaystyle \frac{3}{233}

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Correct Answer

✅ Option c — $\displaystyle \frac{7}{429}$

All Options:

A $\displaystyle \frac{6}{255}$
B $\displaystyle \frac{5}{548}$
C $\displaystyle \frac{7}{429}$
D $\displaystyle \frac{3}{233}$

Detailed Solution & Explanation

**Probability of Two Successive Draws Without Replacement** Total balls = 13 (8 red, 5 white) **Step 1: Probability of drawing 3 white balls in the first draw** - Number of ways to choose 3 white balls from 5:

\binom{5}{3} = 10

- Total ways to choose 3 balls from 13:

\binom{13}{3} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286

P(\text{1st draw is White}) = \frac{10}{286} = \frac{5}{143}

**Step 2: Probability of drawing 3 red balls in the second draw** After the first draw, 10 balls remain (8 red, 2 white). - Number of ways to choose 3 red balls from 8:

\binom{8}{3} = \frac{8 \times 7 \times 6}{6} = 56

- Total ways to choose 3 balls from 10:

\binom{10}{3} = \frac{10 \times 9 \times 8}{6} = 120

P(\text{2nd draw is Red} | \text{1st draw is White}) = \frac{56}{120} = \frac{7}{15}

**Step 3: Joint Probability**

P = \frac{10}{286} \times \frac{56}{120} = \frac{70}{4290} = \frac{7}{429}

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.

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