Negative Binomial Calculator

Input Values



Formula

To calculate the negative binomial:

\[ P = \frac{k \times (1 - p)}{p} \]

Where:

What is a Negative Binomial?

The Negative Binomial is a probability distribution that models the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified number of failures occur. It is characterized by two parameters: the number of failures required, denoted as \( r \), and the probability of success in a single trial, denoted as \( p \).

It helps in calculating the probability of achieving a certain number of successes before encountering a given number of failures.

Example Calculations

Example 1:

Suppose we are flipping a biased coin where the probability of getting a head (success) is \( p = 0.4 \). We want to know how many heads we will get before achieving 3 tails (failures).

Using the formula:

\[ P = \frac{3 \times (1 - 0.4)}{0.4} = \frac{3 \times 0.6}{0.4} = 4.5 \]

The negative binomial value is 4.5.

Example 2:

Consider a situation where the probability of success in a test is \( p = 0.7 \), and we want to determine how many successes will occur before achieving 5 failures.

Using the formula:

\[ P = \frac{5 \times (1 - 0.7)}{0.7} = \frac{5 \times 0.3}{0.7} = \frac{1.5}{0.7} \approx 2.14 \]

The negative binomial value is approximately 2.14.