Related Theory: Statistics and Probability

This problem is a perfect blend of two fundamental pillars of the JEE Mathematics syllabus: Descriptive Statistics (Mean and Variance) and Classical Probability. It requires the ability to solve for unknowns and then apply combinatorial logic.

1. Measures of Dispersion: Variance

Variance measures how far a set of numbers is spread out from their average value. The most used formula in JEE is the computational form:

In competitive exams, always use this form rather than \(\frac{\sum (x_i - \bar{x})^2}{n}\) as it simplifies algebraic manipulation when data points are integers.

2. Solving Symmetric Equations

When you have \(x+y=S\) and \(x^2+y^2=V\), you are essentially solving a quadratic equation. The roots of \(t^2 - (x+y)t + xy = 0\) will give you \(x\) and \(y\). Here, we found \(xy\) using the identity \((x+y)^2\), leading to the quadratic \(t^2 - 14t + 48 = 0\), whose roots are clearly 8 and 6.

3. Probability without Replacement

When picking "one after another without replacement", the order matters (Permutations). However, for problems involving the "minimum" or "maximum" of a set, you can often use Combinations (\(^nC_r\)) because the set of two numbers \(\{2, 3\}\) has the same minimum regardless of whether 2 was picked first or 3 was picked first.

4. The Power of Complementary Events

If a question asks for "at least one", "less than", or "smaller is less than", always check if the opposite case is easier to count. Here, "smaller is less than 4" covers many cases (1, 2, or 3 being picked). The opposite, "both are 4 or greater", is a much smaller subset to calculate.

5. JEE Exam Relevance

Statistics usually provides 1-2 questions per shift in JEE Main. These are considered "easy-scoring" marks. Probability is more varied, but combining it with Statistics is a common tactic to increase the difficulty level from a simple formula-based question to a multi-step logical problem.

6. Common Mistakes

• Calculation Errors: Incorrectly squaring 12 or 14 is the most common pitfall in variance problems.
• Replacement Confusion: Treating the problem as "with replacement" would change the denominator to \(6 \times 6 = 36\).
• Order in Set: Misidentifying \(x-4\) or \(y\) in the second set before solving Step 3.

7. Shortcut: Mean Deviation vs Variance

While Mean Deviation is \(\frac{\sum |x_i - \bar{x}|}{n}\), Variance uses squares to eliminate the sign. Squares penalize outliers more heavily, which is why the Variance in this problem (16) is quite high relative to the mean (8).

8. Properties of Variance

• \(Var(ax + b) = a^2 Var(x)\).
• Variance is always non-negative.
• If all observations are increased by a constant \(k\), variance remains unchanged.

9. Probability Sample Space

For a set of size \(n\), choosing 2 elements without replacement results in \(n(n-1)\) ordered pairs or \(\frac{n(n-1)}{2}\) unordered pairs. Consistency is key—if you use combinations for the numerator, use them for the denominator as well.

10. Summary for Students

To master such problems: 1. Memorize the algebraic relationship between Mean, Variance, and \(\sum x^2\). 2. Practice solving quadratics from sum and product. 3. Always look for the complement in probability to save time.