On this page
Toss a coin and the next toss does not care what came before. Most sequences in the real world are not like that. Whether the next letter in a word is a vowel depends on the letter before it, and tomorrow's weather depends on today's. Andrey Markov showed that the laws of probability can still be made to work when each event depends on the last. This post covers who he was, the argument that pushed him to it, his study of a Russian poem, and a worked example you can do by hand.
Who was Andrey Markov?
His MacTutor biography gives his birth in Ryazan on 14 June 1856 and his death on 20 July 1922. He was in poor health as a child and walked on crutches until he was ten. He entered St Petersburg University in 1874, where Pafnuty Chebyshev's lectures shaped him, won a gold medal for his prize essay in 1878, and earned his doctorate in 1884.
He held firm political views. When the Tsar had Maxim Gorky's election to the Academy withdrawn in 1902, Markov protested. In 1913, while the Romanov dynasty celebrated 300 years in power, he celebrated 200 years of the law of large numbers. In 1917 he asked to be sent to a small town, Zaraisk, where he taught mathematics in the secondary school without pay.
What argument started Markov chains?
The law of large numbers says that the average of many trials settles towards its expected value: toss a fair coin often enough and the fraction of heads approaches one half. Eugene Seneta's paper Markov and the Birth of Chain Dependence Theory explains what Markov was arguing against. The mathematician Nekrasov had claimed that the random quantities in a sum must be pairwise independent for the weak law of large numbers to hold. Seneta's abstract notes that this claim rested on the theological doctrine of free will held by some of the Moscow school.
Markov answered with mathematics. In 1906 he proved such a law for a system of dependent random quantities, in particular for what we now call finite Markov chains. MacTutor describes a Markov chain as a sequence in which the next value is decided by the present one, but not by how the present one came about. The general theory came later; our post on Andrey Kolmogorov covers his 1930s foundations.
How did he test it on a poem?
In 1913 Markov turned to Pushkin's verse novel Eugene Onegin. The English translation of that paper, An Example of Statistical Investigation of the Text Eugene Onegin, opens by describing a sample of 20,000 Russian letters: the whole first chapter and sixteen stanzas of the second. MacTutor notes that he treated text as a chain of two states, vowels and consonants. He counted by hand how often a vowel followed a vowel and how often it followed a consonant.
Worked example: a two-state chain
Sample numbers, not Markov's own: suppose that after a vowel the next letter is a vowel with probability 0.1 and a consonant with 0.9, and after a consonant it is a vowel with probability 0.6 and a consonant with 0.4.
Two letters after a vowel. You can go vowel, vowel, vowel or vowel, consonant, vowel. P = 0.1 × 0.1 + 0.9 × 0.6 = 0.01 + 0.54 = 0.55.
Two letters after a consonant. P = 0.6 × 0.1 + 0.4 × 0.6 = 0.06 + 0.24 = 0.30.
Three letters after a vowel. P = 0.55 × 0.1 + 0.45 × 0.6 = 0.055 + 0.27 = 0.325. After a consonant: 0.30 × 0.1 + 0.70 × 0.6 = 0.03 + 0.42 = 0.45. The two starting points are being pulled together.
The long run. Let v be the long-run share of vowels. A letter is a vowel if the previous one was a vowel and stayed, or was a consonant and switched: v = 0.1v + 0.6(1 − v). So 1.5v = 0.6 and v = 0.4. Check: 0.1 × 0.4 + 0.6 × 0.6 = 0.04 + 0.36 = 0.40.
Why dependence matters. If letters were independent with 40% vowels, two vowels in a row would have probability 0.4 × 0.4 = 0.16. In this chain it is 0.4 × 0.1 = 0.04. The same overall share of vowels hides a very different pattern of pairs, and this is the sort of difference his letter counts could reveal.
What mistakes do students make?
Multiplying as if independent. P(A and B) = P(A) × P(B) only for independent events. For a chain, multiply by the conditional probability of the next step.
Rows that do not add to 1. From each state, the probabilities of all next states must add to 1. Check 0.1 + 0.9 and 0.6 + 0.4 before you start.
"No memory at all." A Markov chain does remember: it remembers the present state.
Expecting the start to matter for ever. Steps 3 and 4 show the start fading as the chain settles near 0.4.
Is this in your syllabus?
Markov chains are not in the JEE (Main) 2026 syllabus, whose probability unit covers events, the addition and multiplication theorems, Bayes' theorem and probability distributions. The skill in steps 1 to 3, multiplying conditional probabilities along each path and adding the paths, is exactly the one tested there; our post on Bayes' theorem and conditional probability practises it.
If you keep multiplying the wrong probabilities together, the Fermi Problem Library has a step-by-step solver that explains the reasoning, so you can see which conditional probability belongs at each step.
What can a student take from Markov's way of working?
Answer a claim with a proof. Markov did not argue about free will; he proved a theorem that settled the mathematical point.
Test ideas on real data. He counted 20,000 letters by hand to see dependence in action.
Hold your principles. From the Gorky protest to unpaid teaching in Zaraisk, he acted on what he believed.
Next step: change the vowel-after-vowel probability in the example to 0.3, keep the rest, and find the new long-run share of vowels.
