Probability Lab

It is the theorem that the empirical frequency of an outcome approaches its theoretical probability as the number of independent trials grows. Flip a fair coin forever and the fraction of heads heads to 0.5. Crucially it says nothing about any single flip and nothing about short runs: it is a statement about long-run averages, which the convergence graph in this lab shows happening in real time.

Theoretical probability comes from counting outcomes: a fair coin has 2 equally likely faces, so P(heads) = 1/2. Empirical probability comes from experiments: flip 100 times, count 47 heads, and the empirical frequency is 0.47. The theoretical value is fixed; the empirical value wobbles around it and settles closer as trials accumulate. The lab shows both at once so the difference is visible rather than abstract.

No. Each flip is independent: the coin has no memory, so the probability of tails stays exactly 1/2. Expecting a correction is the gambler's fallacy. What the law of large numbers actually does is dilute streaks under thousands of newer flips, not cancel them. Run the lab after a visible streak and watch the frequency drift back without any compensating streak of tails.

More than intuition suggests. The wobble of an empirical frequency shrinks roughly with the square root of the trial count, so to halve the noise you need four times the trials. Ten flips can show 0.7 without anything being wrong; a thousand flips lands within a few hundredths of 0.5 in the vast majority of runs. Watch the gap readout: it falls fast at first, then grinds slowly, which is exactly the square-root behavior.

It uses a seeded pseudo-random generator, the same kind games and simulations use. The sequence is statistically fair (every face equally likely, no usable pattern) but reproducible: press Reset and the identical sequence replays, which is deliberate so an experiment can be repeated exactly while still behaving like chance.

For a fair six-sided die, each face has probability 1/6, about 0.167, because six outcomes are equally likely and exactly one counts. The same logic gives 1/n for any fair n-sided object, which is the theoretical line the lab draws. Switch the sides slider to 6, let it run, and watch face 1 settle toward 0.167 while every short stretch of rolls still looks streaky.

Probability compares successes to ALL outcomes; odds compare successes to failures. A fair coin has probability 1/2 of heads but odds of 1 to 1; rolling a 6 has probability 1/6 but odds of 1 to 5. The two encode the same information, so convert with odds = p / (1 - p). Casinos and bookmakers quote odds, textbooks quote probability, and mixing the two is a classic source of wrong answers.

One half times one half: 1/4. Independent events multiply, so a run of k heads has probability (1/2)^k, which is why five in a row (1/32) feels rare but shows up regularly in a few hundred flips. The multiplication rule is also why streaks in the lab are normal rather than evidence of a broken coin: with thousands of trials, improbable runs get plenty of chances to happen.

No. Every trial runs in your browser on the HTML canvas. Nothing is sent anywhere, there is no account, and the lab keeps working offline once loaded.