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Question #41808

Difficulty: 8/10 | 🟢 Free

The Question

Let $a_r$ be the coefficient of $x^r$ in the polynomial $P(x) = \frac{1}{2} \left( (1+x)^{13} + (1-x)^{13} \right)$ . A set of 14 bags, labeled $B_0, B_1, \dots, B_{13}$ , is given. A bag is selected such that the probability of selecting bag $B_r$ is proportional to $a_r$ . Each bag $B_r$ contains $r$ red balls and $13-r$ black balls. A bag is chosen at random according to the given probabilities, and two balls are drawn from it, one after another, with replacement. Given that both drawn balls are red, what is the probability that they were drawn from bag $B_6$ ?

Options

A.
$\frac{297}{896}$
B.
$\frac{297}{1792}$
C.
$\frac{165}{512}$
D.
$\frac{165}{1024}$

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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

Bayes' Theorem Reversal|General Term Formulation|Polynomial Function Parity

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of binomial coefficient properties with Bayes' theorem.

The primary trap lies in the parity of the polynomial $P(x)$ .

Students who mechanically apply the binomial expansion without noting the $+ (1-x)^{13}$ term will assume $a_r = \binom{13}{r}$ for all $r$ .

This leads to an incorrect prior probability denominator of $2^{13}$ instead of $2^{12}$ .

Interestingly, the total probability of drawing two red balls remains $\frac{7}{26}$ in both cases due to the symmetric properties of the second moment of the binomial distribution.

Consequently, missing the parity constraint exactly halves the final conditional probability, leading directly to the trap option $\frac{297}{1792}$ .

Additional traps include calculating for drawing without replacement.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Bayes' Theorem Reversal
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free