Negative Binomial Distribution Calculator

Negative Binomial Formula:

\[ P(r) = C(r+k-1, k-1) p^k (1-p)^r \]

Probability P(r):

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1. What is Negative Binomial Distribution?

The negative binomial distribution describes the probability of having a given number of failures (r) before achieving a specified number of successes (k) in a series of independent Bernoulli trials, each with success probability p.

2. How Does the Calculator Work?

The calculator uses the negative binomial formula:

\[ P(r) = C(r+k-1, k-1) p^k (1-p)^r \]

Where:

\( r \) — Number of failures before k successes occur
\( k \) — Number of successes
\( p \) — Probability of success on each trial
\( C(n, k) \) — Combination (n choose k)

Explanation: The formula calculates the probability of having exactly r failures before the k-th success in a sequence of independent trials.

3. When to Use Negative Binomial Distribution

Details: Use this distribution when you want to know the probability of a certain number of failures before achieving a fixed number of successes, with constant probability of success in each trial.

4. Using the Calculator

Tips: Enter the number of failures (non-negative integer), number of successes (positive integer), and probability of success (between 0 and 1).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between binomial and negative binomial?
A: Binomial counts successes in fixed trials, negative binomial counts failures until fixed successes.

Q2: What are typical applications?
A: Modeling counts like number of sales calls needed for k sales, or disease cases before vaccine success.

Q3: What if k=1?
A: When k=1, it becomes the geometric distribution.

Q4: Can r be zero?
A: Yes, r can be zero (probability of getting k successes immediately).

Q5: What's the expected value?
A: The mean is \( \frac{k(1-p)}{p} \), variance is \( \frac{k(1-p)}{p^2} \).