The Mathematics of Repeated Chances

Each draw on this page is an independent Bernoulli trial with a fixed probability of 16.67%. The sequence of attempts follows the geometric distribution, which describes how many independent trials are needed to observe the first success. The expected number of attempts is the reciprocal of the probability: 1 / (1/6) = 6. This is the average. Individual experiences vary widely.

Cumulative Probability

The chart above plots the probability of winning at least once after N attempts. The formula is P(N) = 1 − (1 − p)^N, where p = 16.67%. After 17 attempts, this cumulative probability reaches 95%. The curve rises steeply at first and then flattens, because each additional attempt adds a smaller marginal probability. Reaching 99% requires roughly 26 attempts, and 99.9% requires approximately 38.

Independence

Every draw on this page is independent of every previous draw. The Web Cryptography API generates fresh randomness for each attempt. Fifty consecutive non-wins do not make the next draw more likely to succeed. This is a core principle of probability theory and the source of the common gambler's fallacy: the mistaken belief that past outcomes influence future independent events. The probability is 16.67% on every single draw, regardless of history.

In the Classroom

This tool makes abstract probability tangible. Set the odds to 1 in 6 (a single die face) and have each student draw until they win. Compare how many attempts each student needed. The class distribution will approximate the geometric distribution, with a long right tail. Students who needed 15 or 20 attempts often feel the result was "unfair," opening a valuable conversation about the difference between expected value and individual experience. For a dramatic demonstration, try 1 in 100 and project the cumulative probability curve. The class can collectively push toward the 95% threshold.

Private by Architecture

Every draw is generated entirely within your browser using the Web Cryptography API with rejection sampling for perfect uniformity. The server delivers the page; your device determines every outcome. Results live in localStorage on your device alone.