How to split the pot of an unfinished game — the letters that birthed probability.
You flip a coin 10 times and get 7 heads. Is something wrong with this coin?
Not for predicting the future — a language for handling uncertainty.
Experiment
Hands-on experiment
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
Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
Flip a coin 10 times and you might get 7 heads. Does that make the coin defective? Short results alone can't tell you.
Probability isn't a tool for guessing 'what happens this time' — it's a language describing 'what structure emerges overall when you repeat'.
Insurance, polling, experiment design, the odds in games — thinking probabilistically is how you judge rationally under uncertainty.
Insight
Insights from the video
“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?'.
Misconception
Common misconceptions
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.
Formula
Writing it as math
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.
In Real Life
Where you meet it in real life
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
Probability is the referee that separates real drug effects from luck. An effect is accepted only when 'the chance of this difference arising by luck is under 5%'.
Try Yourself
Test yourself
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.
Connection
Concepts connect
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
Labs worth exploring together
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 →