Negative Binomial Distribution Calculator

Successes needed

r

Probability

p

Probability type:

Number of failures

x

Upper bound

x_{2}

Output type Chart Table

Decimal Places: 2 3 5 8

Clear Reset

Introduction

The negative binomial distribution models the number of failures $x$ observed before achieving a fixed number of successes $r$ in a sequence of independent trials, each with success probability $p$ . Examining this structure supports the study of waiting-time behaviour, over-dispersed count data, and stochastic processes in which the stopping condition is defined by the attainment of repeated successes.

What this calculator does

The tool performs precise calculations of the Negative Binomial probability mass function and cumulative distribution. Users input the required successes $r$ , the constant probability $p$ , and the failures $x$ . It generates the specific probability, the arithmetic mean $μ$ , and the standard deviation $σ$ . Results are presented as detailed calculation steps and visualised through either a probability distribution chart or a comprehensive data table.

Formula used

The probability mass function is computed using the binomial coefficient for the total trials. The variable $r$ represents the successes, $p$ is the success probability, and $x$ is the number of failures. Log-gamma functions are employed to ensure numerical stability for larger factorials during the calculation of combinations.

P (X = x) = (\binom{x + r - 1}{x}) \cdot p^{r} \cdot {(1 - p)}^{x}

μ = \frac{r (1 - p)}{p}, σ = \sqrt{\frac{r (1 - p)}{p^{2}}}

How to use this calculator

1. Enter the target number of successes $r$ and the probability of success $p$ for a single trial.
2. Select the probability type, such as equal to, less than, or between specific failure counts.
3. Input the number of failures $x$ to be evaluated against the chosen criteria.
4. Execute the calculation to view the resulting probability, mean, and standard deviation alongside the step-by-step breakdown.

Example calculation

Scenario: In a social research study, an investigator monitors trial participants to identify the probability of encountering five non-eligible candidates before successfully recruiting three eligible individuals.

Inputs: $r = 3$ , $p = 0.30$ , and $x = 5$ for $P (X = 5)$ .

Working:

Step 1: $P (X = x) = (\binom{x + r - 1}{x}) \cdot p^{r} \cdot {(1 - p)}^{x}$

Step 2: $(\binom{5 + 3 - 1}{5}) \cdot {0.3}^{3} \cdot {(1 - 0.3)}^{5}$

Step 3: $21 \cdot 0.027 \cdot 0.16807$

Step 4: $0.567 \cdot 0.16807$

Result: 0.0953

Interpretation: There is a 9.53% probability that exactly five failures will occur before the third success is achieved.

Summary: The model successfully quantifies the likelihood of a specific sequence of trial outcomes.

Understanding the result

The output probability reveals the likelihood of a specific number of failures occurring within the defined parameters. The mean $μ$ indicates the average number of failures expected, while the standard deviation $σ$ measures the spread of the distribution around that central value.

Assumptions and limitations

The calculation assumes that trials are independent and the probability $p$ remains constant. The trials must continue until the $r$ -th success is reached, and the variables must be whole numbers within the specified ranges.

Common mistakes to avoid

Users often confuse the number of failures $x$ with the total number of trials $n$ . It is also vital to ensure the success probability is expressed as a decimal between 0 and 1, rather than as a percentage or an integer value.

Sensitivity and robustness

The output is highly sensitive to changes in the probability $p$ , especially when $p$ is very small, as this significantly increases the expected failures. The mean and variance are stable for moderate values but exhibit high volatility as the target success count $r$ increases.

Troubleshooting

If the result returns zero or an error, verify that the input $r$ is a positive integer and $p$ is within the exclusive range of 0 to 1. Ensure the number of failures $x$ does not exceed the maximum computational limit of 5000.

Frequently asked questions

What is the difference between Binomial and Negative Binomial?

In a Binomial distribution, the number of trials is fixed and the number of successes is variable. In a Negative Binomial distribution, the number of successes is fixed and the number of failures or trials is variable.

Can the number of successes be a fraction?

No, the logic of this calculator requires the number of successes $r$ to be a whole number, as it represents a count of discrete events.

What happens if the probability of success is 1?

If the probability is 1, the target successes are achieved immediately with zero failures. Any request for a probability involving more than zero failures will result in zero.

Where this calculation is used

This statistical method is widely applied in probability theory and academic modelling to analyse events that terminate after a specific threshold is met. It is frequently used in population studies to estimate the effort required to find a certain number of individuals with specific traits. In environmental science, researchers use it to model the frequency of specific weather events occurring before a seasonal change. The distribution provides a foundation for understanding over-dispersed data where the variance exceeds the mean, making it a critical tool in advanced discrete data analysis.

Results are based on standard mathematical and statistical methods and may involve rounding or approximation. If precise accuracy is required, please verify results independently. See full disclaimer.