Geometric Distributions

A geometric distribution models the number of trials needed to get the first success in a series of independent Bernoulli trials, where each trial has the same probability of success.

The probability of success (p) is the likelihood of achieving the desired outcome in a single trial.

P(X = k) = (1 - p)^(k-1) * p, where p is the probability of success.

It refers to the scenario where the first success occurs on or after the k-th trial.

P(X >= 3) = 1 - P(X < 3) = 1 - (P(X = 1) + P(X = 2)).

The expected value (mean) is E(X) = 1/p, where p is the probability of success.

The variance is Var(X) = (1 - p) / p^2.

It implies that the probability of making a field goal is 0.9, and the probability of missing is 0.1.

It refers to the trial number at which the first tails appears, which can be modeled using a geometric distribution.

It means there is a 93.75% chance that the first tails will occur within the first 4 flips.