Statistics and Probability

Master measures of central tendency, dispersion, and probability theory for JEE Mathematics.

Statistics deals with data analysis while probability deals with chance and uncertainty.

Overview

graph TD
    A[Stats & Probability] --> B[Statistics]
    A --> C[Probability]
    B --> B1[Central Tendency]
    B --> B2[Dispersion]
    C --> C1[Basic Probability]
    C --> C2[Conditional]

Statistics

Measures of Central Tendency

Mean:

$$\bar{x} = \frac{\sum x_i}{n} = \frac{\sum f_i x_i}{\sum f_i}$$

Median: Middle value when arranged in order

Mode: Most frequent value

Measures of Dispersion

Range: Max - Min

Mean Deviation:

$$MD = \frac{\sum |x_i - \bar{x}|}{n}$$

Variance:

$$\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} = \frac{\sum x_i^2}{n} - \bar{x}^2$$

Standard Deviation:

$$\sigma = \sqrt{\text{Variance}}$$

Coefficient of Variation

$$CV = \frac{\sigma}{\bar{x}} \times 100$$

Used to compare variability of different datasets.

JEE Tip
When data is shifted by constant a: Mean shifts by a, SD unchanged. When data is scaled by constant b: Mean and SD both scale by b.

Probability

Basic Definitions

Sample Space (S): Set of all possible outcomes

Event (E): Subset of sample space

Probability: $P(E) = \frac{n(E)}{n(S)}$

Properties

  1. $0 \leq P(E) \leq 1$
  2. $P(S) = 1$
  3. $P(\phi) = 0$
  4. $P(E') = 1 - P(E)$

Addition Rule

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

For mutually exclusive events:

$$P(A \cup B) = P(A) + P(B)$$

Conditional Probability

$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

Multiplication Rule

$$P(A \cap B) = P(A) \cdot P(B|A) = P(B) \cdot P(A|B)$$

For independent events:

$$P(A \cap B) = P(A) \cdot P(B)$$

Bayes’ Theorem

$$\boxed{P(A_i|B) = \frac{P(B|A_i) P(A_i)}{\sum_{j} P(B|A_j) P(A_j)}}$$

Probability Distributions

Random Variable

A function that assigns numerical values to outcomes.

Expected Value (Mean)

$$E(X) = \sum x_i P(x_i)$$

Variance

$$Var(X) = E(X^2) - [E(X)]^2$$

Binomial Distribution

For n independent trials with probability p of success:

$$P(X = r) = \binom{n}{r} p^r q^{n-r}$$

where q = 1 - p

  • Mean: $\mu = np$
  • Variance: $\sigma^2 = npq$

Practice Problems

  1. Find the variance of: 2, 4, 6, 8, 10

  2. A bag contains 5 red and 3 blue balls. Two balls are drawn. Find probability of both being red.

  3. If P(A) = 0.3, P(B) = 0.4, P(A ∩ B) = 0.2, find P(A|B).

  4. A coin is tossed 6 times. Find probability of exactly 4 heads.

Quick Check
Why do we subtract P(A ∩ B) in the addition rule?

Further Reading