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Question #41080

Difficulty: 8/10 | 🟢 Free

The Question

An urn contains $n$ red balls and $n$ black balls. A random sample of $n$ balls is drawn without replacement. Let $R$ be the number of red balls in the sample. A biased coin, whose probability of showing heads is $\frac{1}{R+1}$ , is then tossed once. Let $P(H)$ be the overall probability of the coin showing heads. If $P(H) = \frac{13}{49}$ , find the variance of $R$ .

Options

A.
$\frac{9}{11}$
B.
$\frac{10}{11}$
C.
$\frac{36}{11}$
D.
$\frac{18}{11}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Multiplication Theorem|Vandermonde's Identity|Binomial Coefficient Summation Properties

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of the Multiplication Theorem (Law of Total Probability) and advanced binomial summations.

The primary trap lies in the evaluation of the expected probability.

Students who fail to recognize the absorption identity $\frac{1}{r+1}\binom{n}{r} = \frac{1}{n+1}\binom{n+1}{r+1}$ will attempt to expand the binomial coefficients into factorials, leading to an intractable algebraic mess that cannot be easily summed.

By successfully applying the identity, the summation transforms into a classic Vandermonde's convolution, collapsing the complex series into a single ratio.

Furthermore, the problem requires knowledge of the variance of a hypergeometric distribution, ensuring that only students with a comprehensive grasp of both combinatorics and probability distributions can reach the final answer.

Quick Facts About This Question

Question Information

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