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Question #40660

Difficulty: 8/10 | 🟢 Free

The Question

Urn $A$ contains $367$ balls, of which $187$ are Red and $180$ are Blue. Urn $B$ contains $367$ balls, of which $180$ are Red and $187$ are Blue. A random variable $K$ follows a hypergeometric distribution, representing the number of Red balls obtained when drawing $187$ balls without replacement from a third Urn $C$ containing $367$ Red and $367$ Blue balls.

Given $K=k$ , two independent experiments are performed: Experiment 1: $k$ balls are drawn without replacement from Urn $A$ . Let $X$ be the number of Red balls drawn. Experiment 2: $187-k$ balls are drawn without replacement from Urn $B$ . Let $Y$ be the number of Red balls drawn.

Find the unconditional probability $P(X + Y = 187)$ .

Options

A.
$\frac{\binom{367}{187}}{\binom{734}{187}}$
B.
$\frac{\binom{367}{180}}{\binom{734}{180}}$
C.
$\frac{\binom{367}{187} - \binom{187}{7}}{\binom{734}{187}}$
D.
$\frac{\binom{360}{180}}{\binom{734}{187}}$

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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

Vandermonde's Identity|Probability Mass Function (PMF)|Complementary Counting|Indicator Function Logic

Why This Question Is Tricky

Common Mistakes

The problem tests the application of the Law of Total Probability combined with Vandermonde's Identity in a compound hypergeometric setting.

The specific trap lies in the complementary counting of 'failure' states: students notice that for $k < 7$ , Urn B does not have enough Red balls ( $187-k > 180$ ), meaning these states are impossible.

They incorrectly assume Vandermonde's Identity $\sum \binom{A}{k}\binom{B}{n-k} = \binom{A+B}{n}$ requires subtracting these 'invalid' boundary cases, forgetting that the standard binomial coefficient $\binom{n}{k}$ is defined as 0 for $k > n$ .

This naturally acts as an indicator function that restricts the summation domain without needing manual normalization, trapping students who overcomplicate the boundary conditions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Vandermonde's Identity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free