Final answer:
The exception refers to a geometric distribution describing the number of trials until the first success in a series of independent trials with constant probability of success, p. The mean of this distribution is 1/p and the variability is based on both p and q=1-p. Unlike a binomial experiment, the number of trials is not fixed.
Step-by-step explanation:
The exception of the number of failures preceding the first success in an infinite series of independent trials with a constant probability of p of success on each trial is described by the geometric distribution.
To define this exception mathematically, we say that if X represents the number of trials until the first success where X can be any whole number starting from 1 (since there must be at least one trial), then X follows a geometric distribution because only the last outcome is a success, indicating that there have been zero or more failures before the success.
The probability, p, of success and the probability, q, of failure do not change from trial to trial, where q = 1 - p. The mean (μ) of the geometric distribution is given by 1/p and the variance by (1-p)/p².
Each trial in this context is called a Bernoulli trial, which has only two possible outcomes: success or failure. This scenario is distinct from a binomial experiment, which has a fixed number of trials. Because in a geometric experiment the number of trials is not fixed and can theoretically continue infinitely, the third characteristic of a binomial experiment is not met.