Statistics and Probability Cheat Sheet - IIT-JEE MAINS 2027

Welcome to the ultimate LibreTexts-style chapter guide for Statistics and Probability, tailored specifically for students preparing for the IIT-JEE MAINS 2027 examination. Mastering these concepts is crucial as Statistics and Probability consistently contribute 2 to 3 high-weightage questions in every NTA JEE Main shift.

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1. Fundamentals of Probability

Probability theory measures the likelihood of events occurring in random experiments. In JEE Main, fundamental questions test foundational axioms, set-theoretic relationships, conditional probabilities, and inverse probability applications using Bayes' Theorem.

Axioms of Probability

Let \(\Omega\) represent the sample space of a random experiment. For any event \(A \subseteq \Omega\):

• Non-negativity & Boundedness: \(0 \le P(A) \le 1\)
• Total Probability of Sample Space: \(P(\Omega) = 1\)
• Countable Additivity: For mutually disjoint events \(A_1, A_2, A_3, \dots\), the probability of their union is \(P\bigl(\bigcup_{i=1}^{\infty} A_i\bigr) = \sum_{i=1}^{\infty} P(A_i)\)

Core Rules & Conditional Probability

Key formulas for set operations and conditional dependencies include:

• Complement Rule: \(P(A^c) = 1 - P(A)\)
• Addition Rule (Inclusion-Exclusion): \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
• Conditional Probability: The probability of event \(A\) given event \(B\) has occurred (where \(P(B) > 0\)):

\[P(A \mid B) = \frac{P(A \cap B)}{P(B)}\]

• Independence Condition: Two events \(A\) and \(B\) are independent if and only if \(P(A \cap B) = P(A)P(B)\).

Law of Total Probability & Bayes' Theorem

If \(B_1, B_2, \dots, B_n\) form a complete partition of the sample space \(\Omega\) such that \(P(B_i) > 0\) and \(B_i \cap B_j = \emptyset\) for \(i eq j\):

Law of Total Probability:

\[P(A) = \sum_{i=1}^n P(A \mid B_i)P(B_i)\]

Bayes' Theorem (Posterior Probability):

\[P(B_k \mid A) = \frac{P(A \mid B_k)P(B_k)}{\sum_{i=1}^n P(A \mid B_i)P(B_i)}\]

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2. Discrete Random Variables & Distributions

A discrete random variable takes on isolatable values. The distribution is defined by its Probability Mass Function (PMF), Expectation, and Variance.

PMF, Expectation, and Variance

• Probability Mass Function (PMF): \(p(x) = P(X = x)\), satisfying \(\sum_{x} p(x) = 1\).
• Expectation (Mean): \(\mathbb{E}[X] = \sum_{x} x \cdot p(x)\)
• Variance: \(\text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2\) where \(\mathbb{E}[X^2] = \sum_{x} x^2 \cdot p(x)\).

Standard Discrete Probability Distributions

1. Bernoulli Distribution: \(\text{Bernoulli}(p)\)

Models a single trial with two outcomes (Success = 1, Failure = 0) with success probability \(p\).

• \(P(X=1) = p\), \(P(X=0) = 1-p\)
• \(\mathbb{E}[X] = p\)
• \(\text{Var}(X) = p(1-p)\)

2. Binomial Distribution: \(\text{Binomial}(n, p)\)

Models the number of successes in \(n\) independent, identical Bernoulli trials.

• PMF: \(P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\) for \(k \in \{0, 1, 2, \dots, n\}\)
• Mean: \(\mathbb{E}[X] = np\)
• Variance: \(\text{Var}(X) = np(1-p)\)

3. Poisson Distribution: \(\text{Poisson}(\lambda)\)

Models the occurrence count of rare events happening at a constant average rate \(\lambda > 0\).

• PMF: \(P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}\) for \(k \in \{0, 1, 2, \dots\}\)
• Mean: \(\mathbb{E}[X] = \lambda\)
• Variance: \(\text{Var}(X) = \lambda\)

4. Geometric Distribution: \(\text{Geometric}(p)\)

Models the trial number \(k\) on which the first success occurs.

• PMF: \(P(X=k) = (1-p)^{k-1}p\) for \(k \in \{1, 2, 3, \dots\}\)
• Mean: \(\mathbb{E}[X] = \frac{1}{p}\)
• Variance: \(\text{Var}(X) = \frac{1-p}{p^2}\)

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3. Continuous Random Variables & Uniform Distribution

Continuous random variables can take any real value within an interval. They are described using Probability Density Functions (PDF) and Cumulative Distribution Functions (CDF).

PDF, CDF, and Expectation

• Probability Density Function (PDF): \(f(x) \ge 0\) such that \(\int_{-\infty}^{\infty} f(x) dx = 1\).
• Cumulative Distribution Function (CDF): \(F(x) = P(X \le x) = \int_{-\infty}^x f(t) dt\)
• Expectation: \(\mathbb{E}[X] = \int_{-\infty}^{\infty} x \cdot f(x) dx\)
• Variance: \(\text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2 = \int_{-\infty}^{\infty} x^2 f(x) dx - (\int_{-\infty}^{\infty} x f(x) dx)^2\)

Continuous Uniform Distribution: \(\text{Uniform}(a, b)\)

A continuous random variable \(X\) is uniformly distributed over the interval \([a, b]\) if the likelihood of landing in any sub-interval depends only on the interval length.

• PDF: \(f(x) = \frac{1}{b-a}\) for \(a \le x \le b\), and \(0\) elsewhere.
• CDF: \(F(x) = \frac{x-a}{b-a}\) for \(x \in [a, b]\).
• Mean: \(\mathbb{E}[X] = \frac{a+b}{2}\)
• Variance: \(\text{Var}(X) = \frac{(b-a)^2}{12}\)

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4. NTA Pattern & PYQ Analysis

According to recent NTA trends for IIT-JEE Mains, Statistics and Probability questions fall into distinct repetitive patterns:

• Binomial Distribution Properties: Solving simultaneous equations given \(\mathbb{E}[X] = np\) and \(\text{Var}(X) = np(1-p)\) to find trial numbers or tail probabilities.
• Conditional Probability with Bayes' Rule: Word problems involving selection from multiple urns, diagnostic testing accuracy, or independent signal transmissions.
• Expectation & Variance Computation: Finding parameters when continuous or discrete functions contain unknown coefficients.

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5. Frequently Asked Questions (FAQ)