Geometric and Hypergeometric Distributions

The geometric distribution is simply the number of tries until your first success, like how many times you roll a die before the first six. It's called geometric because its probabilities form a geometric sequence: each one is the previous one times $(1 - p)$. The hypergeometric distribution is the count of successes when you draw without replacement, like the number of hearts in a 5-card hand; its name comes from a family of series in higher math and won't help you, so just think "drawing without putting back." Neither formula is as bad as it looks: the geometric one is $(1 - p)^{k - 1}$ (fail $k - 1$ times) times $p$ (then succeed), and the hypergeometric one is a fraction of counts, favorable hands over all hands.

Geometric: independent trials with the same $p$, stopping at the first success. It's the binomial setup, but instead of fixing the number of trials and counting successes, you fix the number of successes at one and count trials. Waiting for the first six on a die: $P(\text{first six on roll 3}) = \left(\frac56\right)^2\frac16$. On average it takes $\frac1p = 6$ rolls, which makes sense: if a six comes up a sixth of the time, you'd expect one every 6 rolls. "Within $k$ tries" is $1 - (1 - p)^k$, the complement of failing all $k$.

Hypergeometric: choose $n$ items from $N$ without replacement, where $K$ are successes. Here the binomial breaks, because the draws aren't independent: once you've drawn a heart, there are fewer hearts left. So we go back to counting, as in the probability unit. Choose $k$ of the successes and $n - k$ of the failures, and divide by all ways to choose $n$ items. When the population is huge compared with the sample, removing a few items barely changes the odds, and the hypergeometric is close to the binomial.

A common mistake. After a run of failures, it's tempting to think a success is "due." With independent trials it isn't. The geometric distribution has no memory: after 5 misses, the number of further tries you'll need has exactly the same distribution as when you started.

Geometric: how many sales calls until a sale, how many attempts until a login succeeds. Hypergeometric: card games, lottery odds and acceptance sampling of shipments.