P(A) · Probability
Not for predicting the future — a language for handling uncertainty.
You flip a coin 10 times and get 7 heads. Is something wrong with this coin?
Distinguish equally likely outcomes from one actual observation.
Start by listing possible outcomes and reading simple fractions.
How to split the pot of an unfinished game — the letters that birthed probability.
Open the immersive version →A reading path containing this lesson: Reason under uncertainty · 1/3
What can we say without knowing the next result?
Roll a fair die once. We do not know the next face, but we can list outcomes 1 through 6. Assuming equal likelihood, each face has probability 1/6.
The even outcomes are 2, 4 and 6: three of six equally likely outcomes. Their probability is 3/6=1/2. Equal likelihood, not counting alone, justifies this calculation.
Does that mean exactly one even result in two rolls? No: both could be odd. Probability 1/2 does not command every short run to split exactly in half.
For independent trials with the same probability, we can describe why a large deviation of relative frequency becomes unlikely as the number of trials grows. This still does not determine the next outcome.
A run of heads does not increase the next tail probability for independent flips of a fair coin. If fairness is unknown, observations may instead prompt us to examine that assumption.
Read these symbols together
- Ω
- The sample space: the set of possible outcomes.
- A ⊆ Ω
- An event is a collection of outcomes of interest, such as an even result.
- P(A)
- The probability of event A, not the observed count itself.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
Predict first — you flip a coin 10 times and get 7 heads. Is this coin rigged?
Flip 10 at a time — watch how much the ratio wobbles
Chance is different from an observed count
Short runs wobble; long repetition reveals structure.
Ten coin flips can swing wildly. But flip 1,000 or 10,000 times and the fraction of heads creeps ever closer to 1/2. That's the essence of probability as the law of large numbers states it.
Probability is a story about the whole distribution, not individual events.
Probability answers 'in what proportions do outcomes divide over enough repetitions?' far more clearly than 'what comes up next time?'.
Which outcomes are equally likely?
In the experiment, the ratio approached a fixed value as trials accumulated. Defining that 'destination it approaches' as a formula gives you probability.
Definition of mathematical probability
When every outcome is equally likely, probability is a ratio of outcome counts. Heads is 1 of 2 outcomes, hence 1/2.
Coins and dice
The dashed line (theoretical value) in the experiment's graph was exactly this — the destination the simulation's ratio was heading toward.
The law of large numbers
The fraction kₙ/n of trials where the event occurred converges to the theoretical probability as trials repeat without bound. It's the mathematical name for what you watched happen.
A p-value is interpreted relative to a null hypothesis, model and test statistic.
References
- ASA Statement on p-Values (2016) · Principles 1–3
Make one more distinction
Heads came up 3 times in a row, so tails is 'due' next.
A coin has no memory. Each flip is independent, so the next probability is still 1/2. The feeling of being 'due' is the gambler's fallacy. Long-run balance comes from dilution, not compensation.
If the probability is 1/2, exactly 5 heads in 10 flips is what 'normal' looks like.
The probability of getting exactly 5 in 10 is itself only about 25%. Lopsided short runs are normal — probability is a statement about the structure that emerges over long repetition.
Where else can this idea help?
Insurance pricing
An insurer can't predict any one person's accident. But across a million people, the accident rate emerges stably — the law of large numbers is the mathematical foundation of insurance.
Polls and margins of error
Surveying just 1,000 people can estimate an entire electorate's leanings within ±3 points. It exploits, in reverse, the principle that larger samples bring ratios closer to the truth.
The chance of rain
'70% chance of rain' isn't a claim about predicting tomorrow — it means that among 100 past days with similar conditions, about 70 had rain.
Clinical trials
Under a specified null hypothesis and statistical model, a p-value measures the probability of a result at least as extreme as the observed test statistic. It is not the probability that there is no effect, or that chance caused the data. A 5% threshold alone cannot establish an effect; effect size, uncertainty and study design also matter.
Explain it in your own words
Compare with one possible explanation
Probability assigns a value to an event. A short run allows different sequences, including two odd outcomes.
Try another question
Q1Flip two coins at once. What's the probability both come up heads?Show answer ▾
1/4. The possibilities are (H,H)(H,T)(T,H)(T,T) — four outcomes, one of which works. The common trap is '1/3, since it's one of {two heads, mixed, two tails}' — but 'mixed' is two outcomes.
Q2Roll a die 600 times. Will you get exactly one hundred 6s?Show answer ▾
Almost never. The law of large numbers says 'the ratio approaches 1/6', not 'it lands exactly'. You'll likely be near 100 (90–110), but the probability of exactly 100 is only about 4%.
Q3In the experiment, 10 flips gave 7 heads. If you flip 1,000 times, what happens to the ratio of heads? Will tails come up more often to 'balance it out'?Show answer ▾
No — coins don't pay debts. If later flips come up 1/2 each, the earlier surplus (+2 heads) remains, but gets diluted within the 1,000 (502/1000 = 50.2%). Balance comes from dilution, not compensation.
Try answering yourself before revealing it — getting it wrong is where learning starts.
The idea to take with you
Probability describes chance under a model; it does not guarantee a short-run count or the next result.
What question comes next?
A reading path containing this lesson: Reason under uncertainty · 1/3
Previous concept
Derivatives
The derivative handled the change of definite, certain values. Step out of that world of certainty and the world of probability begins — where outcomes can't be known.
← Derivatives labLeads to next
Functions
Now that you have a language for uncertain situations, it's time for the language that precisely expresses 'one thing depending on another' — functions.
Go to the Functions lab →Related lab
Conditional Probability
The moment you learn something, the probability changes — the Monty Hall problem that mathematicians fought over awaits.
Go to the Conditional Probability lab →Related lab
Expected Value
Multiply probability by money and you get a scale for uncertain choices. Compute a lottery ticket's true price.
Go to the Expected Value lab →