A fair coin lands heads half the time — everyone knows that. So flip one a few times for real. The truth lurches all over the place before it ever settles down. Let's flip hundreds and watch it happen.
Start hereTheoretical probability is what should happen, worked out by careful counting. Experimental probability is what actually happened, tallied from real tries. A handful of tries and they can be miles apart — but pile up enough tries and the gap quietly closes.
A fair coin should come up heads half the time. Yet your first ten flips might be seven heads and three tails — that's experimental probability shrugging and saying "well, this is what I got." Neither flavour is lying. They just need a lot of flips to shake hands.
Probability is just a number between 0 and 1 that measures how likely something is. 0 means "never," 1 means "always," and 0.5 means "fifty-fifty." There are two honest ways to land on that number, and they come at it from opposite directions.
Theoretical probability you work out before you even try, by counting. List every outcome that's equally likely, then ask how many of them count as a "win." A coin has two equally-likely outcomes, and one is heads, so:
A normal six-sided die works the same way: six equally-likely faces, one of them is a four, so P(four) = 1 ÷ 6 ≈ 0.167. No coins were harmed and no dice were rolled — pure counting. That's the "should happen" number.
Experimental probability is the opposite. You actually do the thing, lots of times, and count what really turned up:
Flip a coin 20 times, get 12 heads, and your experimental probability of heads is 12 ÷ 20 = 0.6 — not 0.5, even though the coin is perfectly fair. The whole rest of this page is about that gap: why it shows up, why it's nothing to panic about, and the surprising rule for when it disappears.
Hit a button and a fair coin flips for real. The line tracks your experimental probability of heads after every single flip. The dashed line is the theory: 0.5. Watch the gap.
Theory says 0.5. Start flipping and watch what really happens.
Click Flip once a few times first. The line throws a tantrum — it leaps to 1, crashes toward 0, can't make up its mind. With only a few flips, one lucky head or tail swings the whole proportion wildly. That's experimental probability being noisy: it's reading too few tries to be trusted yet.
Now hammer Flip 100 a few times and keep going. Something almost magical happens. The wild swings on the left shrink to gentle nudges, and the line glides in and hugs the dashed 0.5 line. It never sits perfectly on 0.5 — real life is never that tidy — but it gets close and stays close. The more you flip, the harder it is to escape the theory.
Here's the intuition. After 4 flips, a single extra head moves your proportion by a whole quarter — from 2/4 to 3/4, a giant leap. After 400 flips, one extra head barely budges the number at all: 200/400 versus 201/400 is a change you can hardly see. Each new flip matters less and less to the running total, so the line steadies.
Put it another way: in a short run, a streak of luck can dominate everything. Over a long run, the lucky streaks and unlucky streaks keep cancelling each other out, and what's left underneath is the real 50/50 nature of the coin. The randomness doesn't go away — you just drown it in repetition until the true value rises to the surface.
Mathematicians gave this idea a slightly grand name: the law of large numbers. Don't let the name scare you — it just says what you watched on the chart. As you repeat a random experiment more and more times, the experimental probability closes in on the theoretical probability. Few trials wobble; many trials settle. That's the whole law, and you proved it yourself with a button.
Careful with the word "converge." The line gets closer and closer to 0.5 on average — it doesn't snap onto it and freeze. There's always a tiny shimmer of randomness left. "Closing in," not "locking on."
Say you flip four heads in a row. Your gut screams that tails is now "due" — surely the universe owes you one? It feels obvious. It's also completely wrong, and it's such a famous mistake that it has a name: the gambler's fallacy.
The coin doesn't know what it did last time. It has no memory, no running score to balance, no debt to repay. On flip number five, the chance of tails is exactly what it always was: 0.5. Four heads in a row, ten heads in a row, a hundred — the very next flip is still a clean fifty-fifty. The past flips already happened; they can't reach forward and tip the coin.
"It's landed heads four times — tails is more likely now, it's been building up."
Each flip is independent. After four heads, P(tails) on the next flip is still 0.5. The coin can't remember a streak, so it can't "owe" you anything.
So how does this square with the law of large numbers, which promised things even out? Here's the subtle, beautiful part. The proportion doesn't drift back to 0.5 because tails come along to "make up" for the heads. It drifts back because the early streak gets diluted by sheer volume. Four extra heads is a huge deal out of 8 flips, but a rounding error out of 8,000. The coin never corrects the past — the future just swamps it. Test it on the simulator: flip a streak of heads early, then keep flipping. The line doesn't dip below 0.5 to compensate; it simply drifts back as the pile of later flips grows.
So far theory has been easy: a fair coin is 0.5, a fair die is 1/6, because we could list the equally-likely outcomes. But what if you can't?
Drop a drawing pin on the floor. It can land point-up or point-down on its side. What's the probability of point-up? There's no clean counting argument — the two outcomes aren't equally likely, and the answer depends on the pin's exact shape and weight. Theory just shrugs. The only way to find out is to drop it hundreds of times and count: experimental probability isn't a backup plan here, it's the entire method.
Same story with a bent or weighted coin. Once a coin is squashed out of shape, "heads is 0.5" stops being true, and no amount of counting outcomes will tell you the real number — because the outcomes aren't equally likely anymore. You have to flip it a few hundred times and read the experimental probability off your tally. That number becomes your best estimate of how that particular weird coin behaves.
Lopsided objects with no "fair" symmetry. The only way to know the odds is to throw them many times and count.
Forecasters can't list equally-likely weathers. They look at how often days like today turned rainy in the past — experimental probability, on a huge scale.
No formula says how likely a goal is. You count goals ÷ shots over a whole season to estimate it.
Designers run a level thousands of times to measure how often players win — then tune the difficulty.
This is the real grown-up reason experimental probability matters. Theory is gorgeous when the world is tidy and symmetrical. The moment it isn't, experiment quietly takes over — and the law of large numbers is what lets us trust it, as long as we gather enough tries.
Each face should turn up 1 time in 6 — about 0.167. The pale bars are that theory; the solid bars are what you actually rolled. Roll a lot and watch them line up.
Each face should come up about 0.167 of the time. Roll and watch the bars find that level.
With a dozen rolls the bars look ragged — one face shoots up, another flatlines, and it's tempting to think the die is "hot" or "cold." It isn't. That's just the noise of too few rolls, exactly like the coin's early wobble. Keep rolling into the hundreds and the six solid bars all sink toward the same pale 1/6 level. Six little experimental probabilities, all closing in on their shared theoretical value at once. The law of large numbers doesn't care whether it's two outcomes or six.
No marks, no pressure — just see whether the ideas have clicked. Tap your answer.
That's the gambler's fallacy beaten. Each flip is independent, so the past five heads change nothing. The next flip is a clean fifty-fifty: P(tails) = 0.5. The proportion evens out later by being diluted, not by the coin "owing" you tails.
Few trials wobble. Experimental probability is just times-it-happened ÷ total tries = 9 ÷ 30 = 0.3. Theory predicts 0.167, but 30 rolls is far too few to settle — a gap like this is exactly the noise the law of large numbers smooths away over hundreds of rolls.
When symmetry breaks, experiment takes over. A bent coin's outcomes aren't equally likely, so counting can't give you 0.5 anymore. And two flips are pure noise. You have to gather a big pile of tries and read the experimental probability — that's your best estimate of how this particular coin behaves.
Theory = what should happen, from counting. Experiment = what did happen, from tries.
Few tries wobble wildly; many tries settle toward the theory. That's the law of large numbers.
A fair coin owes you nothing. The past evens out by being swamped, never by "correcting" itself.