ProbabilityPYQ Jan 26Question 4291 of 187
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If two dice are rolled, then the probability of getting a greater number on the first die than the one on the second, given that the sum should be equal to 7 is

Options

A12\displaystyle \frac{1}{2}
B16\displaystyle \frac{1}{6}
C13\displaystyle \frac{1}{3}
D23\displaystyle \frac{2}{3}
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Correct Answer

✅ Option a — 12\displaystyle \frac{1}{2}

All Options:

  • A12\displaystyle \frac{1}{2}
  • B16\displaystyle \frac{1}{6}
  • C13\displaystyle \frac{1}{3}
  • D23\displaystyle \frac{2}{3}

Detailed Solution & Explanation

Let (x,y)\displaystyle (x, y) represent the outcome when two dice are rolled, where x\displaystyle x is the number on the first die and y\displaystyle y is the number on the second die.
We are given the condition that the sum of the numbers is 7\displaystyle 7. Let B\displaystyle B be the event that the sum is 7\displaystyle 7: B={(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)}  ⟹  n(B)=6B = \{(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)\} \implies n(B) = 6
Let A\displaystyle A be the event that the first die shows a greater number than the second die (x>y\displaystyle x > y). Within the given event B\displaystyle B, the outcomes satisfying x>y\displaystyle x > y are: - (4,3)\displaystyle (4, 3) - (5,2)\displaystyle (5, 2) - (6,1)\displaystyle (6, 1) So, the number of favorable outcomes is: n(A∩B)=3n(A \cap B) = 3
The conditional probability of A\displaystyle A given B\displaystyle B is: P(A∣B)=n(A∩B)n(B)=36=12P(A | B) = \frac{n(A \cap B)}{n(B)} = \frac{3}{6} = \frac{1}{2} Hence, **Option A** 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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