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Probability can feel like a bag of formulas: one for "not", one for "or", one for "at least". In fact almost all of them follow from three short rules. Those rules were set down by Andrey Kolmogorov in 1933, and they are in your Class 11 textbook. This post covers who he was, what the axioms say, and how to use them to solve and check problems.
Who was Andrey Kolmogorov?
Kolmogorov was born in Tambov, Russia, on 25 April 1903 and died in Moscow on 20 October 1987, according to his MacTutor biography. His mother died at his birth, and he was brought up by his aunt. After school he worked for a while as a conductor on the railway and, in his spare time, wrote a treatise on Newton's laws of mechanics.
He entered Moscow State University in 1920 and studied metallurgy and Russian history as well as mathematics. While still an undergraduate he constructed a function that surprised the experts and made his name known internationally. He graduated in 1925, the year his first paper on probability appeared, written with Aleksandr Khinchin, and became a professor at Moscow University in 1931.
His later work ranged from turbulence in fluids to the motion of planets. He also gave a great deal of time to a school for gifted children, planning syllabuses, writing textbooks and teaching. Among many honours, he was elected to the Indian Statistical Institute in 1962 and the Royal Society of London in 1964.
What did Kolmogorov's 1933 book do?
His monograph Grundbegriffe der Wahrscheinlichkeitsrechnung ("Basic Concepts of Probability Theory"), published in 1933, built probability theory rigorously from axioms, in a way MacTutor compares with Euclid's treatment of geometry. It answered the probability part of Hilbert's sixth problem, which asked for axioms for the physical sciences in which mathematics plays an important part.
Glenn Shafer and Vladimir Vovk (2006) describe the book as having "put probability's modern mathematical formalism in place", while also offering an explanation of how the formalism connects to experience. The NCERT Class 12 probability chapter says in its historical note that the book introduced probability as a set function.
What are the axioms in your textbook?
The NCERT Class 11 Probability chapter, section 14.2, defines probability P as a function whose domain is the power set of the sample space S and whose values lie between 0 and 1, satisfying:
For any event E, P(E) ≥ 0.
P(S) = 1.
If E and F are mutually exclusive, P(E ∪ F) = P(E) + P(F).
That is all. Everything else is a consequence.
Worked example 1: derive two rules you already use
P(not E) = 1 − P(E). E and E′ are mutually exclusive, and together they make up S. By axiom 3, P(E) + P(E′) = P(S), and by axiom 2 that is 1. So P(E′) = 1 − P(E).
P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Split A ∪ B into A and "B but not A". These are mutually exclusive, so P(A ∪ B) = P(A) + P(B − A). Also B is "A and B" together with "B but not A", so P(B) = P(A ∩ B) + P(B − A). Subtract to get the formula.
Worked example 2: is this a valid assignment?
A die is weighted so that P(1) = P(2) = P(3) = 0.1, P(4) = P(5) = 0.2 and P(6) = 0.3.
Every value is at least 0, so axiom 1 holds.
The total is 0.1 + 0.1 + 0.1 + 0.2 + 0.2 + 0.3 = 1.0, so axiom 2 holds.
Then P(even) = P(2) + P(4) + P(6) = 0.1 + 0.2 + 0.3 = 0.6, by axiom 3.
If a question gave each face 0.2, the total would be 1.2, which breaks axiom 2. No such die exists.
Worked example 3: a JEE-style bound
Question. P(A) = 0.7 and P(B) = 0.6. What is the smallest possible value of P(A ∩ B)?
Since A ∪ B is inside S, P(A ∪ B) ≤ 1. So 0.7 + 0.6 − P(A ∩ B) ≤ 1, which gives P(A ∩ B) ≥ 0.3. In particular A and B cannot be mutually exclusive, because then P(A ∪ B) would be 1.3.
Common mistakes students make
Adding probabilities of events that overlap. Axiom 3 needs mutually exclusive events. For overlapping events, subtract P(A ∩ B).
Mixing up "mutually exclusive" and "independent". Exclusive events cannot happen together. Independent events do not affect each other. Two events with non-zero probability cannot be both.
Answers above 1. A probability of 1.2 is a sign that something was counted twice.
Forgetting the sample space. Write S down. It tells you what "1" refers to.
Our guide to common probability mistakes has more traps of this kind, and the post on Jacob Bernoulli and the law of large numbers shows the older, frequency side of the subject. When you are not sure which rule to apply, the Fermi Problem Library has a step-by-step solver that explains the reasoning rather than printing an answer.
What can a student take from Kolmogorov's way of working?
Find the few rules underneath. He reduced a whole subject to a handful of axioms. Try writing each probability formula you know as a consequence of the three axioms.
Range widely. History, metallurgy, turbulence and planetary motion all fed his thinking. Interests outside your syllabus are not wasted time.
Give your knowledge away. He spent years teaching schoolchildren. Explaining to others is part of how deep understanding is built.
Write the three axioms on a card today and, for each probability question you solve this week, note which axiom each step uses.
