NIMCET Probability & Statistics: Bayes' Theorem, Distributions & Standard Deviation

Probability & Statistics is a high-yield scoring module in NIMCET Mathematics, contributing 5 to 8 questions (60 to 96 marks). Questions range from classical card/dice problems to multi-stage Bayes' Theorem computations and statistical variance calculations.

Chapter Architecture & Weightage Matrix

PROBABILITY & STATISTICS BLUEPRINT
+-----------------------------------------------------------------------------------+
| 1. CLASSICAL PROBABILITY | 2. PROBABILITY LAWS     | 3. STATISTICS & DISTRIBUTIONS|
| - Permutation/Combination | - Total Probability Thm | - Mean, Median, Mode         |
| - Independent Events     | - Bayes' Theorem        | - Variance & Std Deviation   |
| - Conditional $P(A|B)$   | - Random Variables $E(X)$| - Binomial Distribution $B(n,p)$|
+-----------------------------------------------------------------------------------+

Exam Weightage

Core Probability & Statistics Formulas

1. Bayes' Theorem

$P(E_i | A) = \frac{P(E_i) \cdot P(A | E_i)}{\sum_{j=1}^n P(E_j) \cdot P(A | E_j)}$

2. Binomial Distribution Parameters

For random variable $X \sim B(n, p)$:

• Mean $\mu = E(X) = n p$
• Variance $\sigma^2 = \text{Var}(X) = n p q \quad (\text{where } q = 1 - p)$

3. Variance Shortcut Formulas

• Variance $\sigma^2 = \frac{\sum x_i^2}{N} - (\bar{x})^2$
• Variance of first $n$ natural numbers: $\sigma^2 = \frac{n^2 - 1}{12}$

Solved NIMCET Past Year Questions

Question 1 (Bayes' Theorem)

A bag contains 4 red and 6 black balls. Another bag contains 6 red and 4 black balls. A bag is chosen at random and a ball is drawn. If the drawn ball is red, find the probability that it came from the first bag:

• (A) $\frac{2}{5}$
• (B) $\frac{4}{10}$
• (C) $\frac{2}{5}$
• (D) $\frac{2}{5}$

• Solution:
  1. $P(B_1) = P(B_2) = \frac{1}{2}$.
  2. $P(R | B_1) = \frac{4}{10} = \frac{2}{5}$, and $P(R | B_2) = \frac{6}{10} = \frac{3}{5}$.
  3. By Bayes' Theorem: $P(B_1 | R) = \frac{\frac{1}{2} \times \frac{4}{10}}{\left(\frac{1}{2} \times \frac{4}{10}\right) + \left(\frac{1}{2} \times \frac{6}{10}\right)} = \frac{4}{4 + 6} = \frac{4}{10} = \frac{2}{5}$
• Correct Option: (A)

Question 2 (Variance of Natural Numbers)

The standard deviation of the first 11 natural numbers is:

• (A) $\sqrt{10}$
• (B) 10
• (C) $\sqrt{12}$
• (D) 5

• Solution:
  1. Formula for variance of first $n$ natural numbers: $\sigma^2 = \frac{n^2 - 1}{12}$.
  2. For $n = 11$: $\sigma^2 = \frac{11^2 - 1}{12} = \frac{120}{12} = 10$.
  3. Standard Deviation $\sigma = \sqrt{10}$.
• Correct Option: (A)

Frequently Asked Questions (FAQ)

Q1: Is Poisson Distribution required for NIMCET?

While Binomial Distribution is far more frequent, basic knowledge of Poisson probability mass function $P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}$ is recommended.