Statistics · A module in nine lessons
Probability, from zero to exam level
Probability and distributions account for 335 questions in our bank of past Economics entrance papers, from one-line definitions in CUET to the Delhi School’s hardest puzzles. This module teaches the subject from the ground up, one idea at a time, until those questions feel routine.
344past-paper questions, each placed in a lesson
114of them from the Delhi School, gathered in Lesson 9
9lessons, all live
0prerequisites beyond school algebra and fractions
This is a course, not a single article.9 lessons that build on each other, each about 40 minutes, with its own interactives, checked exercises and graded past papers. Take them in order, or let the diagnostic tell you where to begin.
The path
The lessons
0 of 9 doneYour progress is saved on this device.
- Lesson 1
Sample spaces and counting
What a probability measures, why two dice have 36 outcomes and not 11, and the counting rules that turn most questions into a fraction.
- Lesson 2
The rules of probability
Events as sets, the three axioms and what follows from them, the addition rule for two and three events, and the bounds on an overlap.
- Lesson 3
Conditional probability and independence
Conditioning shrinks the sample space. Trees, total probability, independence and why it is not the same as mutually exclusive.
- Lesson 4
Bayes' theorem
Turning P(evidence | cause) into P(cause | evidence). Natural frequencies, the base-rate trap, and every Bayes pattern the papers use.
- Lesson 5
Random variables, expectation and variance
A random variable as a function, pmf and CDF, expectation as a balance point, variance, and the indicator trick.
- Lesson 6
Discrete distributions
Bernoulli, binomial, geometric, hypergeometric and Poisson: when each applies, and their means and variances.
- Lesson 7
Continuous distributions
Density as area, uniform, exponential and normal, and transformations of a random variable.
- Lesson 8
Sums of random variables, and the inequalities
The mean and variance of a weighted sum, portfolio risk and diversification, sums that stay normal or Poisson, Markov, Chebyshev and the law of large numbers, and Jensen's inequality.
- Lesson 9
The hard problems
The Delhi School's hardest questions, sorted by technique: first-step analysis, Markov chains and random walks, contests, optimal stopping.
Two halves
Lessons 1 to 4 are about events: counting, the rules, conditioning and Bayes. Lessons 5 to 7 are about random variables: numbers that depend on chance, their averages and spreads, and the standard distributions. Lesson 9 gathers the Delhi School’s hardest questions and the techniques that crack them. Each part builds on the one before, so take them in order.
Where to start
A five-minute diagnostic
Fourteen questions, two for each lesson so far. Your answers tell you which lesson to start with.
Question 1 of 14
Two fair dice are rolled. What is the probability that the sum is 7?
The goal
What you will be able to solve
Nine real questions, from the easiest level to the hardest. By the end of the module every one of them should take you under three minutes.
FoundationCUET 2025 · Lesson 1
If you toss a balanced coin twice, what is the probability of at least one head?
FoundationJNU 2017 · Lesson 2
Two events are said to be mutually exclusive if…
FoundationHyderabad 2011 · Lesson 6
What is the mean of a binomial distribution with n = 20 and p = 0.5?
StandardJNU 2018 · Lesson 3
Box 1 has 3 red and 5 white balls, box 2 has 4 red and 2 white. A ball moves from box 1 to box 2, then one is drawn. P(white)?
StandardJNU 2014 · Lesson 4
A and B tell the truth with probabilities 0.6 and 0.8. If they make the same statement, what is the chance it is true?
StandardJAM 2024 · Lesson 7
X is exponential with mean 50. Find P(X > 70 | X > 60).
AdvancedDSE 2014 · Lesson 1
Two restaurants are located at random on a 10 km street. What is the chance they are less than 5 km apart?
AdvancedDSE 2004 · Lesson 4
Three boxes each have two drawers: gold–gold, silver–silver, gold–silver. A drawer shows gold. P(the box is gold–gold)?
AdvancedDSE 2014 · Lesson 5
5 men and 5 women sit at random in a row. What is the expected number of women sitting next to at least one man?
How it teaches
How each lesson works
- 1A question to hold on to
Each lesson opens with a real exam problem you can't yet solve, and solves it at the end.
- 2Intuition, then picture, then formula
The formula arrives last, as shorthand for something you already understand.
- 3Pause and predict
Commit to an answer before reading on. Being wrong first is what makes it stick.
- 4Worked exam problems
Real past questions, reworked step by step at three levels.
- 5The traps
The wrong options examiners actually use, and why they tempt.
- 6Check, then practise
Checked exercises, a one-screen summary, then graded past papers.
The evidence
Where the questions come from
Delhi School of Economics130
JNU102
Hyderabad36
CUET-PG34
IIT JAM23
UGC NET7
Questions available for practice on these two topics, by exam.
The Delhi School of Economics asks the most, and the hardest: expectation tricks, Markov chains and Bayes puzzles. JNU asks the most often at a standard level. CUET and UGC NET keep to definitions and one-step problems, but ask them every year.
So the module starts gently and goes all the way. Lessons 1 to 7 end with foundation and standard sets at the level of today’s exams; every DSE question above the easy level waits in Lesson 9, sorted by technique.
Real questions
Practise across lessons 1–9
Foundation
Definitions and one-step problems. Most CUET and UGC NET questions sit here.
117 questionsPractise →
Standard
Two or three steps, up to the hardest JAM, JNU and Hyderabad questions.
113 questionsPractise →
Advanced
The Delhi School of Economics' hardest questions, collected in Lesson 8.
114 questionsLesson 9 →
These are questions from the TestStop bank. A free account answers 10 a day, each with its worked solution; All-Access opens every set with no daily limit.
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