Negative Binomial

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 occurs, then $X$ is a discrete random variable called a negative binomial.

Negative Binomial Distribution

Consider the following probability mass function:

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

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 , where $q=1-p$.

• is the negative binomial probability, meaning the probability of having $x-1$ successes after $n-1$ trials and having $x$ successes after $n$ trials.

Note: Recall that $\binom{n}{x}=\frac{n!}{x!(n-x)!}$. For further review, see the Combinations and Permutations Tutorial.

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 independent Bernoulli trials, so for each $i$ (where $1\leq i \leq n$):

$X_i = \begin{cases} 1, & \text{if the } i^{th}\text{ trail is a success} \\ 0, & \text{otherwise } \end{cases}$

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

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

Example

Bob is a high school basketball player. He is a $70\%$ free throw shooter, meaning his probability of making a free throw is $0.70$. 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, $\newline g(n=5, p=0.7) = 0.3^4 0.7 = 0.00567$

# The probability that a machine produces a defective product 
# is 1/3. What is the probability that the 1st defect is found 
# during the 5th inspection?
#
# Answer 0.066
print(round((((1-ratio)**4)*ratio), 3))

# What is the probability that the 1st defect is found during 
# the first 5th inspections?
def geo_dist(i, r):
    return ((1-r)**i)*r

r = 1/3
t = 5
print(round(sum([geo_dist(i, r) for i in range(0, t)]), 3))