Geometric distribution

Negative Binomial Experiment

A negative binomial experiment is a statistical experiment that has the following properties:

The experiment consists of n repeated trials.
The trials are independent.
The outcome of each trial is either success (s) or failure (f).
$P(s)$ is the same for every trial.
The experiment continues until x successes are observed

If $X$ is the number of experiments until the $x^{th}$ success occures, then $X$ is a discrete random variable called a negative binomial

Negative Binomial Distribution

Consider the following probability mass function:

$$b^*(x,n,p) = {\binom{n-1}{x-1}}p^xq^{n-x}$$

The function above is negative binomial and has the following properties:

The number of successes to be observed is $x$.
The total number of trials is $n$.
The probability of success of 1 trial is $p$.
The probability of failure of 1 trial $q$, where $q=1-p$.
$b^*(x,n,p)$ is the negative binomial probability, meaning the probability of having exactly $x-1$ successes out of $n-1$ trials and having $x$ successes after $n$ trials.

Geometric Distribution

The geometric distribution is a special case of the negative binomial distribution that deals with the number of Bernoulli trials required to get a success (i.e., counting the number of failures before the first success). Recall that $X$ is the number of successes in $n$ independent Bernoulli trials, so for each $i$ (where $1\leq i\leq n):

$ X_i =

\begin{cases} 1 if the i^{th} trial is a success \\ 0 otherwise x \end{cases}

The geometric distribution is a negative binomial distribution where the number of successes is 1. We express this with the following formula:

$$g(n,p)=q^{n-1}p$$

Example

Bob is a high school basketball player. He is a 70% free throw shooter, meaning his probability of making a free throw is 0.7. What is the probability that Bob makes his first free throw on his fifth shot?

For this experiment n=5, p=0.7 and q=0.3 So :

$$g(n=5, p=0.7)=0.3^4 0.7=0.00567$$