Binomial Distribution vs Geometric Distribution

A binomial setting has a fixed number of trials, two outcomes per trial, independent trials, and the same success probability each time. For example, the number of heads in 10 flips of a fair coin is binomial with $n = 10$ trials and success probability $p = 0.5$. The probability of exactly $k$ successes is $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ (n choose k, times p to the power k, times one minus p to the power n minus k). Its mean is $\mu = np$ (n times p).

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.