Negative Binomial Probability Calculator

Negative Binomial Distribution Formula:

\[ P(X = k) = C(k-1, r-1) \times p^r \times (1-p)^{k-r} \]

Number of Trials (k):

trials

Required Successes (r):

successes

Success Probability (p):

probability

Probability P(X = k):

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1. What is Negative Binomial Distribution?

The negative binomial distribution models the number of trials needed to achieve a specified number of successes in a sequence of independent Bernoulli trials. It's useful for modeling count data where the number of successes is fixed but the number of trials varies.

2. How Does the Calculator Work?

The calculator uses the negative binomial probability formula:

\[ P(X = k) = C(k-1, r-1) \times p^r \times (1-p)^{k-r} \]

Where:

\( P(X = k) \) — Probability of exactly k trials until r successes
\( C(k-1, r-1) \) — Combination of k-1 choose r-1
\( p \) — Probability of success in each trial
\( r \) — Required number of successes
\( k \) — Total number of trials

Explanation: The formula calculates the probability that exactly k trials are needed to achieve r successes, with the last trial always being a success.

3. Applications of Negative Binomial Distribution

Details: Commonly used in quality control, epidemiology, ecology, and business analytics. Examples include modeling the number of sales calls needed to achieve a certain number of sales, or the number of patients needed to find a specific number with a rare disease.

4. Using the Calculator

Tips: Enter the total number of trials (k), required number of successes (r), and probability of success per trial (p between 0 and 1). Ensure k ≥ r and r > 0 for valid calculations.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between binomial and negative binomial?
A: Binomial distribution counts successes in fixed trials, while negative binomial counts trials needed for fixed successes.

Q2: When is negative binomial preferred over Poisson?
A: When data shows overdispersion (variance > mean), negative binomial often provides better fit than Poisson.

Q3: What are typical parameter ranges?
A: k ≥ r > 0, 0 ≤ p ≤ 1. The mean is r/p and variance is r(1-p)/p².

Q4: Can p be exactly 0 or 1?
A: p=0 gives probability 0 (impossible success), p=1 gives probability 1 if k=r, 0 otherwise.

Q5: How is this related to geometric distribution?
A: Geometric distribution is a special case of negative binomial where r=1 (first success).