# Geometric Distribution

A geometric distribution is a discrete probability distribution of a random variable. Suppose I am in a mood to dance in a party hall and i randomly ask people to dance. Now let X denote the number of people i ask in order to dance. Suppose now, if the first person accepts then X = 1. If the first person refuses and the second person accepts then the value of X will be 2 and it goes on. Now When X = n, where i failed for the first (n - 1) times and succeeded on the nth try. The probability of failure for the first try will be (1 - p), and for the first two tries it is (1 - p)(1 - p). So now the probability of failing for first n - 1 tries will be $(1-p)^{n-1}$ and the probability of succeeding at the nth try is p.

Therefore **P(X = n) = $(1-p)^{n-1}$ p** This is known as Geometric distribution. **A geometric distribution is a discrete probability distribution of a random variable X which satisfies the following conditions:**

1. A trial is repeated until a success occurs.

2. The trials will be independent and the probability will be constant for each trial.

3. Many things will be repeated until a success occurs. For example you may take your driver's exam several times before you pass and acquire your driver's license.