Probability: why the numbers settle down the more times you try

How to use this simulation

1. Start on "Coin flip" and press the roll button once — it shows heads or tails, and both the count on the left and the bar on the right grow together.
2. Press Play to run many flips, and raise the rolls-per-second slider to go faster. Watch the experimental bar creep steadily toward the theoretical 0.5 line.
3. Switch to "Sum of two dice" and change the target sum — the grid lights up every one of the 36 cells that adds up to it.
4. In "Drawing balls from a bag", change the red/blue/green counts and watch the theoretical (outlined) bars jump instantly, even before a single ball has been drawn.
5. After many trials, watch the "Average gap from theory" reading — it keeps shrinking, which is the law of large numbers happening in front of you.

What happens if you flip a coin a hundred times

Flip a coin once and there is no way to say for certain whether it lands heads or tails. Flip it a hundred times, though, and you can confidently predict close to 50 heads — and flip it ten thousand times and that prediction gets sharper still. Even a genuinely random event settles into a pattern once the numbers get big enough, and that pattern is exactly what probability studies.

Casinos, insurance companies and weather services all run on this one fact. Nobody can say what happens to a single customer, but the average behaviour of millions of them can be predicted with remarkable precision. Press Play in the simulation above and watch, with your own eyes, how something genuinely random still drifts toward a fixed number.

We use the word "probably" constantly in everyday speech without ever pinning it to a number — "it will probably rain today", "there's a good chance of passing the exam", "that team is unlikely to win". Mathematical probability takes that same everyday intuition and ties it to a precise number, so two different chances can actually be compared instead of just being called "more" or "less" likely.

From zero: trials, outcomes and the sample space

A random experiment is any process whose result cannot be predicted for certain in advance, even though every possible result is known ahead of time — flipping a coin or rolling a die, for instance. Each possible result is called an outcome, and the complete set of every outcome is the sample space, written S. A coin's sample space is S = {H, T}; a die's is S = {1, 2, 3, 4, 5, 6}.

Any subset of the sample space that we care about is called an event (E) — "rolling an even number" is an event, containing the outcomes {2, 4, 6}. The outcomes that make an event true are its favourable outcomes.

If every outcome in the sample space is equally likely to happen — a fair coin or a fair die, for example — the outcomes are called equally likely, and the simplest formula applies directly: favourable outcomes divided by total outcomes.

Probability always sits between 0 and 1. A value of 0 means the event is impossible; 1 means it is certain. It can never come out negative, and it can never come out bigger than 1 — if a calculation lands outside that range, something in the working has gone wrong.

The vocabulary, before the formulas

A quick glossary before the formulas below.

The formulas: from theoretical to experimental

The classical formula for equally likely outcomes carries most of the weight in this topic. Two more rules and one real-world definition round it out.

Try this in the simulation

Work through each of these before the solved problems below.

Solved problems

Nine problems, spanning a coin, a die, two dice, a bag of balls and real recorded data. Every number here is computed by the page's own code.

Common mistakes

Most marks lost in probability come from miscounting or misreading wording, not from misunderstanding the idea.

• Applying the simple formula to outcomes that are not equally likely — assuming every face of a biased die, or every side of a lopsided coin, has the same chance.
• Miscounting the sample space — treating (1,2) and (2,1) as the same outcome when rolling two dice, when they are two distinct outcomes.
• Confusing drawing with replacement and drawing without replacement — always check first whether the total shrinks on the second draw.
• Writing a probability bigger than 1, as if it were a percentage without dividing by 100, or writing a fraction upside down.
• Forgetting the complement rule and trying to recount "not happening" from scratch, when 1 − P(E) already gives the answer.
• Applying the independent-events multiplication rule to events that are actually dependent, such as two draws from a bag without replacement.

Real-life uses

Probability quietly drives decisions well beyond the exam hall.

• Weather forecasting: a "70% chance of rain" means that on roughly 70% of past days with the same atmospheric conditions, it rained — a direct application of experimental probability.
• Insurance: premiums are set from statistics on how long people of a given age typically live.
• Factory quality control: testing a sample of thousands of items to estimate the defect rate of an entire batch — exactly bonus problem 9.
• Medicine: how likely a drug is to work is measured by trialling it across groups of patients.
• Genetics: the chance a child inherits a particular eye colour or blood type from their parents' genes is a probability calculation.
• Games and gambling: every casino game and lottery is designed so that, over the long run, the odds favour the house by exactly the law of large numbers.

Exam corner

On NCERT/CBSE boards (class 10, "Probability"), expect direct classical-probability questions on a coin, die or a deck of cards, plus a longer, multi-part question built around a bag of coloured objects or two dice.

Later, in class 11 ("Probability" again, with a wider toolkit), the same sample-space thinking extends to permutations and combinations for counting larger sample spaces — get comfortable counting n(S) and n(E) by hand now.

Full marks need the sample space S and the event E written out explicitly, with n(S) and n(E) clearly shown, before the final fraction — a bare final answer with no working loses the method marks even when it is numerically correct.

One-screen revision

A last look before the exam.

Frequently asked questions