Negative Binomial Calculator
Instantly calculate probability mass, cumulative distribution, mean, and variance for Negative Binomial distributions.
| Failures (k) | P(X = k) | P(X ≤ k) |
|---|
* Table shows values around your target k and the expected mean.
What is a Negative Binomial Calculator?
A negative binomial calculator is a specialized statistical tool designed to compute probabilities related to the negative binomial distribution. Unlike a standard binomial calculator (which fixes the number of trials), the negative binomial distribution models the number of failures ($k$) that occur in a sequence of independent Bernoulli trials before a specified number of successes ($r$) is achieved.
This calculator is essential for quality control engineers, data scientists, and researchers who need to understand “stopping time” problems—scenarios where the experiment continues until a fixed goal is met. It helps answer questions like, “What is the probability that I will fail 10 times before I finally make 3 sales?”
Common misconceptions include confusing this with the geometric distribution (which is a special case where $r=1$) or the standard binomial distribution (where the number of trials is fixed, not the successes).
Negative Binomial Calculator Formula and Explanation
The mathematical foundation of the negative binomial calculator is the Probability Mass Function (PMF). The formula for the probability of observing exactly $k$ failures before achieving $r$ successes is:
Where C(n, k) represents the combinations formula (n choose k).
Variable Definitions
| Variable | Meaning | Typical Range | Role in Calculator |
|---|---|---|---|
| r | Target number of successes | Integer ≥ 1 | Determines when the experiment stops. |
| p | Probability of success | 0 < p < 1 | Likelihood of success in a single trial. |
| k | Number of failures | Integer ≥ 0 | The variable outcome we measure probability for. |
| 1 – p | Probability of failure | 0 < (1-p) < 1 | Likelihood of failure in a single trial. |
Practical Examples
Example 1: The Persistent Salesperson
Scenario: A real estate agent needs to sell 3 houses ($r=3$) to reach their monthly bonus quota. Historically, they close a sale with 10% of clients ($p=0.10$). They want to know the probability that they will face exactly 20 rejections ($k=20$) before hitting their 3rd sale.
Input into Negative Binomial Calculator:
- Number of Successes (r): 3
- Probability of Success (p): 0.1
- Target Failures (k): 20
Result: The probability is approximately 0.038 (3.8%). The mean number of failures expected would be $r(1-p)/p = 3(0.9)/0.1 = 27$ failures.
Example 2: Server Uptime Checks
Scenario: A network admin is testing a flaky connection. A “success” is a successfully transmitted packet. The connection is considered stable once 50 packets transmit successfully ($r=50$). The packet success rate is high at 95% ($p=0.95$). What is the risk that we see 5 or more dropped packets ($k=5$) before finishing?
Calculation: Using the calculator, we would check the Cumulative Distribution Function (CDF) for $k=4$ and subtract from 1, or sum P(X=k) for k=5 to infinity.
Mean Failures: $50(0.05)/0.95 \approx 2.63$ dropped packets expected.
How to Use This Negative Binomial Calculator
- Identify Your Goal (r): Enter the total number of successes you require to stop the experiment. This must be a whole number (e.g., 5 sales, 3 heads, 10 passed tests).
- Determine Probability (p): Input the probability of success for a single, independent trial. This comes from historical data or theoretical probability (e.g., 0.5 for a coin flip).
- Set Target Failures (k): Enter the specific number of failures you are investigating.
- Analyze Results:
- P(X=k): The exact chance of having that many failures.
- Cumulative P(X≤k): The chance of having that many failures OR FEWER.
- Mean: The average number of failures you should expect in the long run.
Key Factors That Affect Negative Binomial Results
- Success Probability (p): As $p$ decreases, the expected number of failures increases drastically. If tasks are harder, you will fail more often before succeeding $r$ times.
- Target Successes (r): Increasing $r$ increases the mean number of failures linearly. Waiting for 10 successes takes twice as long (on average) as waiting for 5.
- Variance Sensitivity: A lower $p$ not only increases the mean failures but significantly increases the variance ($r(1-p)/p^2$). This means results become much more volatile and unpredictable.
- Independence Assumption: The calculator assumes trials are independent. If a failure makes the next failure more likely (momentum or fatigue), the negative binomial model will underestimate risk.
- Sample Size: For very large $r$, the negative binomial distribution approaches a Normal distribution, making the “bell curve” shape visible in the chart.
- Interpretation of Failures vs. Trials: Be careful with terminology. This calculator counts failures ($k$). Total trials ($n$) would be $k + r$. Always clarify if your data source counts total attempts or just the failed ones.
Frequently Asked Questions (FAQ)
In a Binomial distribution, the number of trials is fixed (e.g., flip a coin 10 times) and the number of successes varies. In a Negative Binomial distribution, the number of successes is fixed (e.g., flip until 3 heads) and the number of trials/failures varies.
Yes. The Geometric distribution is simply a Negative Binomial distribution where $r=1$ (stopping after the first success). Set “Number of Successes” to 1.
The distribution usually peaks near the mean. Outcomes significantly higher than the average number of failures become increasingly unlikely as the probability density tapers off.
If $p=1$, you never fail, so $k$ is always 0. If $p=0$, you never succeed, so the experiment never ends (undefined mean). The calculator restricts inputs to $0 < p < 1$.
No. In this calculator, $k$ refers strictly to the number of failures. The total number of trials would be $k + r$.
Yes, it is often used in credit risk modeling to estimate the number of default-free accounts (failures) processed before observing a certain number of defaults (successes) to calibrate risk models.
Mathematically, Variance = Mean / p. Since $0 < p < 1$, dividing the mean by a decimal increases the value. This property (overdispersion) distinguishes it from the Poisson distribution where Mean = Variance.
The calculator uses double-precision floating-point arithmetic. It is accurate for most practical engineering and statistical applications, though extreme inputs (e.g., r > 1000) may encounter minor precision limits.