Geometric distribution

A discrete probability distribution for the number of independent trials with constant success probability needed to obtain the first success, or the failures preceding it.

Bernoulli trial: A random experiment with two outcomes, commonly called success and failure. The distribution models the waiting time across repeated Bernoulli trials with a fixed success chance.
Independent and identically distributed random variables: Random variables that are mutually independent and share the same probability distribution. Repeated trials meet this condition when each has the same success chance and does not affect the others.
Negative binomial distribution: A discrete distribution for the number of trials needed to obtain a specified number of successes. It generalizes geometric waiting by allowing more than one success to be required.
Coupon collector's problem: A probability problem about how many random draws are needed to collect every type in a set. Its total collection time is a sum of geometric waiting times with changing success probabilities.
Probability mass function: A function assigning probabilities to the possible values of a discrete random variable. Its formula gives the probability of each possible waiting time.
Binomial distribution: The distribution of the number of successes in a fixed number of independent Bernoulli trials. It counts successes across a fixed horizon, while the geometric distribution counts trials until the first.
Poisson distribution: A discrete distribution for the number of events in a fixed interval under a constant-rate model. It counts events over a fixed interval rather than trials until a first success.
Reliability engineering: The discipline of assessing and improving the dependability of systems and components. A geometric model describes the number of independent demands a component survives before failure.
Memoryless property: A property whereby the remaining wait has the same distribution regardless of elapsed time. The geometric distribution is the only discrete waiting-time distribution with this property.
Conditional probability: The probability of an event given that another event has occurred. Conditioning on earlier failures makes the geometric distribution's memorylessness precise.

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