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Question #41709

Difficulty: 8/10 | 🟢 Free

The Question

15 identical coins are distributed into 3 distinct boxes. Let $x_i$ be the number of coins in the $i$ -th box ( $i=1,2,3$ ). Let $E$ be the event that $\sum_{i=1}^3 \left[ \frac{x_i}{4} \right] = 2$ , where $[ \cdot ]$ denotes the greatest integer function. Let $F$ be the event that $x_i \in [0, 7)$ for all $i=1,2,3$ . The conditional probability $P(F|E)$ is

Options

A.
$\frac{1}{12}$
B.
$\frac{1}{2}$
C.
$\frac{1}{9}$
D.
$\frac{2}{9}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Conditional Probability Constraints|Distributing Identical Items (Stars and Bars)|Greatest Integer Function Traps|Closed vs Open Intervals

Why This Question Is Tricky

Common Mistakes

This problem tests the intersection of combinatorics (Stars and Bars) and the Greatest Integer Function within a conditional probability space.

The primary trap lies in the 'Closed vs Open' interval constraint $x_i \in [0, 7)$ .

Students frequently misinterpret this as $x_i \le 7$ instead of the strict integer inequality $x_i \le 6$ .

This end-point trap leads to an incorrect numerator of 18 (yielding the trap answer 1/2).

Additionally, a secondary trap exists if students fail to restrict the remainder $r_i \le 3$ when calculating the denominator, leading to an inflated sample space and incorrect probabilities like 1/9 or 2/9.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Conditional Probability Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free