This tool calculates the probability and cumulative distribution for the negative binomial distribution based on the number of successes and probability of success.

## How to Use the Negative Binomial Distribution Calculator

This calculator helps you compute the probability of a given number of successes in a series of trials, given a specific probability of success. Here’s how to use it:

- Enter the number of successes (k) you are interested in.
- Enter the probability of success (p) for each trial (as a decimal between 0 and 1).
- Enter the number of failures (r) you want to account for.
- Click the “Calculate” button to get the result.

## Explanation of Calculation

The negative binomial distribution is used to find the probability of a specified number of successes occurring in a fixed number of trials. The formula used is:

`P(X = k) = C(k + r - 1, k - 1) * p`

^{k} * (1 - p)^{r}

Where:

**k**is the number of successes.**p**is the probability of success on an individual trial.**r**is the number of failures.**C(n, k)**is the combination function, calculated as`n! / (k! * (n - k)!)`

.

## Limitations

The Negative Binomial Distribution Calculator has some limitations:

- The probability of success (p) should be between 0 and 1.
- The number of successes (k) should be a non-negative integer.
- The number of failures (r) should be a non-negative integer.

## Use Cases for This Calculator

### Calculate the Probability of Achieving a Specific Number of Failures Before a Given Number of Successes

Enter the probability of success, the number of successes needed, and the number of failures you want to calculate. The calculator will provide you with the probability of achieving that exact number of failures before reaching the specified number of successes.

### Determine the Probability of Reaching a Target Number of Successes in a Fixed Number of Trials

Input the probability of success and the target number of successes desired in a set number of trials. Once you hit calculate, the tool will show you the probability of achieving the defined number of successes within the specified trials.

### Estimate the Probability of Obtaining Success After a Certain Number of Failures

Specify the probability of success and the number of failures. The tool will give you the probability of achieving success after the stated number of failures, allowing you to make informed decisions based on the calculated outcome.

### Compute the Cumulative Probability of Successes Accumulated Over Multiple Trials

Provide the success probability, the number of successes, and the number of trials that have been conducted. Upon calculation, the tool will display the cumulative probability of achieving the specified number of successes across the given trials.

### Find the Probability of Experiencing a Certain Number of Failures Before a Success

Input the success probability and the number of failures required before a success. Upon running the calculation, you will discover the likelihood of encountering the exact number of failures specified before achieving success.

### Determine the Probability of Achieving a Certain Number of Successes Within a Range of Trials

Enter the success probability, the target number of successes, and the range of trials. The calculator will provide you with the likelihood of achieving the defined number of successes within the given range of trials.

### Calculate the Expected Number of Failures Before Obtaining a Fixed Number of Successes

Specify the success probability and the number of successes. The tool will compute the expected number of failures anticipated before reaching the defined number of successes, offering insights for your decision-making process.

### Estimate the Probability of Success in x Trials, Given a Fixed Success Count

Provide the success probability and the desired number of successes. The calculator will reveal the probability of achieving the specified number of successes within a fixed number of trials, assisting you in gauging your likelihood of success.

### Find the Probability of Encountering a Certain Number of Failures Before Reaching Success in x Trials

Input the success probability, the number of trials, and the target number of failures before success. The tool will indicate the likelihood of encountering the specified number of failures before achieving success across the designated number of trials.

### Compute the Probability of Achieving Success Within a Certain Number of Trials

Enter the success probability and the maximum number of trials allowed. Upon calculation, the tool will display the probability of achieving success within the specified number of trials, enabling you to set realistic expectations and plan accordingly.