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Question #41076

Difficulty: 8/10 | 🟢 Free

The Question

A number is chosen at random from the set $S = \{1, 2, 3, \dots, 1000\}$ . Given that the chosen number is divisible by $2$ or $3$ , what is the probability that it is neither divisible by $4$ nor by $5$ ?

Options

A.
$\frac{334}{667}$
B.
$\frac{333}{667}$
C.
$\frac{350}{667}$
D.
$\frac{297}{667}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Conditional Probability Foundations|Complementary Counting|Cardinality Inclusion-Exclusion

Why This Question Is Tricky

Common Mistakes

The problem tests the intersection of compound divisibility events using Cardinality Inclusion-Exclusion and Complementary Counting.

The primary trap lies in the expansion of $(D_2 \cup D_3) \cap (D_4 \cup D_5)$ .

Students often fail to recognize that $D_{12}$ is a subset of $D_4$ , leading to massive overcounting if they blindly apply the inclusion-exclusion principle on four sets.

Furthermore, calculating the intersections requires correctly identifying the Least Common Multiple (LCM) of the divisors (e.g., $D_4 \cap D_{10} = D_{20}$ , not $D_{40}$ ), which is a classic pitfall in combinatorial probability.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Conditional Probability Foundations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free