To calculate the negative binomial:
\[ P = \frac{k \times (1 - p)}{p} \]
Where:
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 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.
