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Question #41827

Difficulty: 8/10 | 🟢 Free

The Question

The mean and variance of 12 observations are 4 and $\frac{7}{3}$ , respectively. Later, it was found that two observations, 1 and 7, were incorrect. After removing these two incorrect observations, the corrected data is found to consist of only two distinct integer values. If 4 observations are drawn randomly without replacement from the corrected data, what is the probability that the variance of this sample of 4 observations is strictly less than 1?

Options

A.
$\frac{11}{21}$
B.
$\frac{10}{21}$
C.
$\frac{6}{11}$
D.
$\frac{5}{11}$

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

Complementary Counting|Sampling With/Without Replacement|Correction of Incorrect Mean/Variance

Why This Question Is Tricky

Common Mistakes

The question tests the student's ability to correct statistical parameters and seamlessly transition into combinatorial probability.

The primary trap lies in the correction phase: students often forget to decrement the sample size ( $n$ ) from 12 to 10 after purging the incorrect data points, leading to an incorrect variance and frequency distribution.

A secondary trap is calculating the probability of all valid variance cases directly instead of using complementary counting ( $1 - P(\text{Var} = 1)$ ), which is significantly faster since the maximum possible variance of a sample drawn from this binary dataset occurs when the sample is perfectly balanced.

Quick Facts About This Question

Question Information

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