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Question #41796

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{1, 2, 3, \dots, 1500\}$ . A number $n$ is chosen uniformly at random from $S$ . Let $E$ be the event that $\sum_{k=1}^{n} (302 - |k - 302|) \ge 0$ Find the probability of the event $E$ .

Options

A.
$\frac{103}{150}$
B.
$\frac{1207}{1500}$
C.
$\frac{1031}{1500}$
D.
$\frac{343}{500}$

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

Sample Space Partitioning|Sigma Notation Properties|Compound Function Boundaries

Why This Question Is Tricky

Common Mistakes

The question tests the student's ability to correctly partition a sample space based on a summation involving a piecewise function (absolute value).

The core trap lies at the boundary $k=302$ .

Students often fail to split the summation $\sum_{1}^{n}$ into $\sum_{1}^{302} + \sum_{303}^{n}$ , instead incorrectly applying the algebraic form $604-k$ to the entire domain, which leads to the trap answer 1207/1500.

Another common error is approximating the discrete summation with a continuous definite integral, which yields the trap answer 1031/1500.

Quick Facts About This Question

Question Information

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