In a binomial setting, the number of trials n is fixed in advance, and the binomial random variable X counts the number of successes. The possible values of X are 0, 1, 2… n. In other situations, the goal is to repeat a chance process until a success occurs.
These are all examples of a geometric setting. Although the number of trials isn't fixed in advance, the trials are independent and the probability of success remains constant.
A geometric setting arises when we perform independent trials of the same chance process and record the number trials it takes to get one success. On each trial, the probability p of success must be the same.
In a geometric setting, if we define the random variable Y to be the number of trials needed to get the first success, then Y is called a geometric random variable. The probability distribution of Y is a geometric distribution.
The number of trials Y that it takes to get a success in a geometric setting is a geometric random variable. The probability of Y is a geometric distribution with parameter p, the probability of a success on any trial. The possible values of Y are 1,2,3…
Geometric Probability Formula
P(Y=k) = p(1-p)^(k-1)
When using a calculator, you can use geometpdf(p,k) -> P(Y = k) or geometcdf(p,k) -> P(Y ≤ k)
If it is < or >, subtract one from your x-value to account for the equal issue.
Mean (Expected Value) of a Geometric Random Variable
If Y is a geometric random variable with probability of success p on each trial, then its mean is µy = E(Y) = 1/p. That is, the expected number of trials required to get the first success is 1/p.