ProbabilityPYQ May 25Question 4383 of 187
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Two cards are drawn from a pack of 52 cards. The probability that one is a spade and one is a heart; is

Options

A320\displaystyle \frac{3}{20}
B2934\displaystyle \frac{29}{34}
C47100\displaystyle \frac{47}{100}
D13102\displaystyle \frac{13}{102}
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Correct Answer

✅ Option d — 13102\displaystyle \frac{13}{102}

All Options:

  • A320\displaystyle \frac{3}{20}
  • B2934\displaystyle \frac{29}{34}
  • C47100\displaystyle \frac{47}{100}
  • D13102\displaystyle \frac{13}{102}

Detailed Solution & Explanation

A standard pack contains 52\displaystyle 52 cards, where:
- Number of spade cards = 13\displaystyle 13
- Number of heart cards = 13\displaystyle 13

Two cards are drawn from the pack. The total number of ways to draw 2 cards from 52 is:
N(S)=(522)=52×512×1=26×51=1326N(S) = \binom{52}{2} = \frac{52 \times 51}{2 \times 1} = 26 \times 51 = 1326
We want to find the probability that one card is a spade and the other is a heart. The number of favorable outcomes is the number of ways to choose 1 spade out of 13 and 1 heart out of 13:
n(E)=(131)×(131)=13×13=169n(E) = \binom{13}{1} \times \binom{13}{1} = 13 \times 13 = 169
The probability is:
P(E)=n(E)N(S)=1691326P(E) = \frac{n(E)}{N(S)} = \frac{169}{1326} Dividing both the numerator and the denominator by 13:
P(E)=13102P(E) = \frac{13}{102}
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.

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