On this page
Toss a coin ten times and you might get seven heads. Toss it a thousand times and you will almost never be far from half. Everyone feels this is true. Proving it, and saying how many tosses are enough, was the work of a Swiss mathematician more than 300 years ago. This post covers Jacob Bernoulli's life, what his theorem says, and how to see it for yourself with the combinations you learn in Class 11.
Who was Jacob Bernoulli?
Jacob Bernoulli was born in Basel on 6 January 1655 and died there on 16 August 1705, according to his MacTutor biography. His parents made him study philosophy and theology; he took degrees in both, and studied mathematics and astronomy against their wishes. He became professor of mathematics at Basel in 1687 and held the chair until his death.
With his younger brother Johann he worked through Leibniz's new calculus, which most mathematicians then found obscure. MacTutor notes that Jacob's 1690 paper on the isochrone is the first place the word "integral" appears with its present meaning. The brothers later became bitter rivals, which MacTutor describes in some detail. Jacob also asked for a logarithmic spiral to be carved on his tombstone, with the words "Eadem mutata resurgo", meaning "I shall arise the same though changed".
What is Ars Conjectandi?
His book on probability, Ars Conjectandi ("The Art of Conjecturing"), was published in Basel in 1713, eight years after his death. In a history written for its 300th anniversary, Eugene Seneta (2013) explains that the fourth part, which contains the theorem, was left unfinished. Jacob's nephew Nicolaus wrote in the preface that illness and early death had stopped Jacob from completing it, and chose to publish it as it stood.
What does the law of large numbers say?
Seneta describes two sides to Bernoulli's theorem:
The limit. As the number of independent trials grows, the fraction of trials in which an event happens gets closer to its probability. This is what we now call the weak law of large numbers.
A sample size. Bernoulli did not stop at "eventually". He gave a number of trials large enough to make it, say, a thousand times more likely than not that the observed fraction is within a chosen distance of the true probability.
His own example was an urn with 30 "fertile" and 20 "sterile" items, so the probability of drawing a fertile one is 3/5. He asked how many draws, with replacement, make it at least 1000 times more likely than not that the fraction of fertile draws is within 1/50 of 3/5. Seneta reports that later, sharper approximations put the answer at about 6,500 draws.
A worked example with a fair coin
The binomial distribution is no longer a section of the rationalised NCERT Class 12 probability chapter, and the JEE (Main) 2026 syllabus does not list it, but you can do this with nCr alone.
Question. You toss a fair coin 10 times. What is the probability that between 40% and 60% of the tosses are heads?
40% to 60% of 10 tosses means 4, 5 or 6 heads.
The number of ways to get exactly k heads is 10Ck. So 10C4 = 210, 10C5 = 252 and 10C6 = 210.
Favourable outcomes: 210 + 252 + 210 = 672.
Total outcomes: 2¹⁰ = 1024.
Probability = 672/1024 ≈ 0.656.
So about one time in three, ten tosses land outside 40-60% heads. Repeat the sum for 100 tosses, adding 100Ck for k = 40 to 60 (a spreadsheet does this quickly), and the probability rises to about 0.965. For 1,000 tosses it is greater than 0.9999999. The band stays the same, 40-60%, but landing inside it becomes almost certain. That is Bernoulli's theorem, seen in numbers.
Common mistakes students make
The gambler's fallacy. After five tails in a row, the next toss is still 50-50. The law of large numbers works by swamping early streaks with many later tosses, not by correcting them.
Mixing up the fraction and the count. In 1,000 tosses, 510 heads is a fraction of 0.51, very close to 0.5, even though it is 10 heads more than half.
Expecting small samples to behave. Ten tosses missing the 40-60% band about a third of the time is normal, not a sign of a biased coin.
Forgetting independence. The theorem assumes each trial does not affect the next. Drawing cards without putting them back breaks that.
For more traps in probability questions, see our guide to common probability mistakes. If the counting step is what trips you up, 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 Bernoulli's way of working?
Ask "how many?" as well as "does it?" Bernoulli was not satisfied that frequencies settle down eventually. He wanted a number. When you learn a result, ask what it says about a specific case.
Test an idea with a model you can count. His urn of 30 and 20 items is a model you can count exactly, the same habit as using coins and dice in practice questions.
Unfinished work can still matter. Ars Conjectandi was incomplete, yet it shaped probability for centuries. A rough solution written down is worth more than a perfect one left in your head.
Tonight, toss a coin 20 times, record the running fraction of heads after each toss, and plot it to watch it settle.
