Geometric Distribution vs Poisson 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.

The Poisson distribution counts how many times something happens in a fixed window of time or space, assuming events arrive independently at a steady average rate $\lambda$ (lambda). If a help desk averages 3 calls per hour, the probability of exactly 2 calls in an hour is $\frac{3^2 e^{-3}}{2!} = \frac{9(0.0498)}{2} \approx 0.224$. The general formula is $P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$, and both the mean and the variance equal $\lambda$. The Poisson distribution is not part of AP Statistics at all, though it is standard in college probability and statistics courses.