Geometric distribution

The geometric distribution asks how long you wait for a first success, so its possible values are 1, 2, 3, and onward with no upper limit. Rolling a fair die until the first 6 has $p = 1/6$, and the probability the first 6 arrives on roll 3 is $(5/6)^2(1/6) = 25/216 \approx 0.1157$. In general $P(X = k) = (1-p)^{k-1}p$, and the mean is $\mu_X = 1/p$ (mu-X, the mean of X), which here is 6 rolls. The geometric distribution was removed from the AP Statistics exam in the Fall 2026 redesign, so treat it as standard college material rather than tested content.