Geometric Distribution (First Success)

To have your first success on trial k you must fail k − 1 times and then succeed, which is $(1-p)^{k-1} \times p$. Rolling a fair die, the chance the first six comes on the third roll is (5/6)² × (1/6) = 25/216 ≈ 0.116. Enter p as a decimal from 0 to 1 or switch that input to %; k is a whole trial number, 1 or more.

The geometric distribution is memoryless, the only discrete distribution that is, so after twenty failed rolls the chance the next roll is a six is still 1/6, exactly as it was at the start. That is the gambler's fallacy in its purest form, and the "hot hand" is its mirror image. Notice that the single most likely value of k is always 1, however small p is: each additional wait multiplies by (1 − p) < 1. Solving for p has no closed form, so the calculator finds it numerically, the smaller of the two that fit; k comes straight out: with p = 0.2 and P = 0.08192, k = 1 + ln(0.4096)/ln(0.8) = 5.