ProbabilityMCQMTP June 2023 Series IIQuestion 2828 of 295
All Questions

From a deck of 52\displaystyle 52 cards, two cards are drawn at random. What is the probability that they are a king and a queen, if the cards are drawn one after the other without replacement?

Options

A452×451\displaystyle \frac{4}{52} \times \frac{4}{51}
B452×451+452×451\displaystyle \frac{4}{52} \times \frac{4}{51} + \frac{4}{52} \times \frac{4}{51}
C2×4×352×51\displaystyle \frac{2 \times 4 \times 3}{52 \times 51}
D4×452×51\displaystyle \frac{4 \times 4}{52 \times 51}
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Correct Answer

✅ Option b — 452×451+452×451\displaystyle \frac{4}{52} \times \frac{4}{51} + \frac{4}{52} \times \frac{4}{51}

All Options:

  • A452×451\displaystyle \frac{4}{52} \times \frac{4}{51}
  • B452×451+452×451\displaystyle \frac{4}{52} \times \frac{4}{51} + \frac{4}{52} \times \frac{4}{51}
  • C2×4×352×51\displaystyle \frac{2 \times 4 \times 3}{52 \times 51}
  • D4×452×51\displaystyle \frac{4 \times 4}{52 \times 51}

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Detailed Solution & Explanation

To find the probability that the two cards drawn without replacement are a king and a queen, we consider two mutually exclusive cases: 1. Drawing a King first and then a Queen: - The probability of drawing a King first is 452\displaystyle \frac{4}{52} (since there are 4 kings in 52 cards). - The probability of drawing a Queen second, without replacement, is 451\displaystyle \frac{4}{51} (since there are 4 queens left in 51 cards). - Probability of this order is 452×451\displaystyle \frac{4}{52} \times \frac{4}{51}. 2. Drawing a Queen first and then a King: - The probability of drawing a Queen first is 452\displaystyle \frac{4}{52} (since there are 4 queens in 52 cards). - The probability of drawing a King second, without replacement, is 451\displaystyle \frac{4}{51} (since there are 4 kings left in 51 cards). - Probability of this order is 452×451\displaystyle \frac{4}{52} \times \frac{4}{51}. Total probability is the sum of the probabilities of these two cases: Total Probability=452×451+452×451\displaystyle \text{Total Probability} = \frac{4}{52} \times \frac{4}{51} + \frac{4}{52} \times \frac{4}{51}. Hence, **Option B** 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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