Before You Begin

Question #41803

Difficulty: 8/10 | 🟢 Free

The Question

Let $X$ be a discrete random variable taking values in the set of non-negative integers $\mathbb{W} = \{0, 1, 2, \dots\}$ . The probability mass function of $X$ is given by $P(X=k) = C \binom{k+2}{2} t^k$ where $t = \frac{1}{\pi} \sin^{-1}\left(\sin \frac{8\pi}{9}\right)$ and $C$ is a positive constant such that $\sum_{k=0}^{\infty} P(X=k) = 1$ . Find the conditional probability $P(X=1 \mid X \le 2)$ .

Options

A.
$\frac{9}{38}$
B.
$\frac{72}{227}$
C.
$\frac{9}{91}$
D.
$\frac{72}{217}$

Check Your Answer

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 Constraints|Binomial Theorem for Negative/Fractional Indices|Composition Inverse Identity Traps

Why This Question Is Tricky

Common Mistakes

The question tests the student's vigilance with inverse trigonometric composition identities, specifically $f^{-1}(f(x)) = x$ , which is only valid when $x$ is within the principal range of $f^{-1}$ .

Here, $\sin^{-1}(\sin \frac{8\pi}{9})$ is often incorrectly evaluated as $\frac{8\pi}{9}$ by ignoring the $[-\frac{\pi}{2}, \frac{\pi}{2}]$ range restriction.

Because the incorrect value $t = \frac{8}{9}$ is strictly less than 1, the resulting infinite binomial series still converges, masking the error and leading to a valid but incorrect probability mass function.

The conditional probability structure ensures the normalizing constant $C$ cancels out, isolating the penalty purely to the domain trap and the binomial coefficient expansion.

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