geometric distribution
The geometric distribution answers a waiting question: how many tries until the first success? Keep rolling a die until you get a six — how many rolls? Keep selling policies until one lapses — how many policies? The count of attempts up to and including the first success (or the number of failures before it) follows a geometric distribution. It is the distribution of patience.
It has one parameter p, the success probability on each independent trial. The probability that the first success comes on trial number k is (1-p)^(k-1) · p — you need k-1 failures and then a success. The mean number of trials is 1/p (if success has chance 1 in 5, you wait 5 trials on average) and the variance is (1-p)/p^2. Its signature property is memorylessness: having already waited many failures does not change how much longer you must wait — the process has no memory of the past. It is the discrete cousin of the exponential distribution.
Actuaries meet the geometric whenever they model time-to-event in whole-number steps: the number of years a policy stays in force before lapse, the number of periods until a claim, the run length before a regime changes. In a Bonus-Malus or no-claims-discount system the number of claim-free years is geometric-flavoured. Its memoryless property is a strong, sometimes unrealistic, assumption — in reality lapse and claim rates often change with policy age, so the plain geometric is a starting model rather than a final one.
If a policy lapses with probability p = 0.1 each year (independently), the number of years until it lapses is geometric with mean 1/0.1 = 10 years. But memorylessness means a policy that has lasted 8 years is still expected to last another 10 — often unrealistic.
The geometric models 'how long until the first event' — but assumes the odds never change.
Beware two common conventions: 'number of trials including the success' (support 1, 2, 3, ...) versus 'number of failures before the success' (support 0, 1, 2, ...) — they differ in mean by exactly 1, so always state which you mean.