Run random experiments and watch the empirical results settle toward the exact theory โ the law of large numbers, made visible. Every draw uses your browser's cryptographic randomness. Nothing leaves your device.
What you're seeing: each flip is independent. The running proportion of heads jumps around wildly at first, then homes in on the true chance p as trials pile up. That settling is the law of large numbers.
What you're seeing: the sum of the dice. Sums in the middle can happen many ways, so they're far more likely than the extremes. The exact curve (amber outline) comes from convolving the dice โ every combination counted precisely.
| Sum | Exact P | Observed P | Ways |
|---|
Set outcomes and their weights. Bigger weight = bigger slice = more likely.
What you're seeing: observed frequencies (blue) climbing toward each slice's expected frequency (amber). The chi-square number below is an informal "how far from expected are we?" gauge โ it shrinks toward its typical value as you spin more.
Spin to compute a chi-square value.
What you're seeing: draw a card, ask "did the event happen?", repeat. With replacement the deck is reshuffled every draw, so each draw is independent. Without, cards are removed until the deck runs out, then reshuffled. Either way a single draw's probability is the event's fraction of the 52 cards.
What you're seeing: draw a handful and count the target color. Without replacement the exact answer is the hypergeometric distribution; with replacement it's the binomial. The histogram shows how often each count shows up vs. that exact distribution.
How likely is it that at least two people share a birthday? (365 days, no leap years.)
What you're seeing: the famous surprise โ with just 23 people the chance of a shared birthday is already about 50.7%. Each simulated "room" fills with random birthdays; a red dot marks anyone who collides. The empirical estimate converges to the exact formula curve.