Negative Binomial Distribution Calculator
This calculator computes the probability of achieving a specified number of failures before achieving a certain number of successes, based on the probability of success for a single trial. Input the values to calculate and visualize the distribution.
P(X = ):
P(X < ):
P(X ≤ ):
P(X > ):
P(X ≥ ):
Understanding the Negative Binomial Distribution
The Negative Binomial distribution represents the probability of observing a certain number of failures before a fixed number of successes, given a constant probability of success on each trial. It's widely used in scenarios where repeated independent trials are conducted, such as quality control or sports analytics.
Key Components of the Negative Binomial Distribution
- Probability of Success (p): The probability of achieving success on a single trial.
- Number of Successes (r): The number of successful outcomes required.
- Number of Failures (k): The number of failures observed before the required successes are achieved.
Negative Binomial Formula
The probability mass function (PMF) for the Negative Binomial distribution is given by the formula:
Example Calculation
Suppose we want to calculate the probability of observing exactly 4 failures before achieving 3 successes, with a success probability of 0.5.
- \( p = 0.5 \) (probability of success)
- \( k = 4 \) (number of failures)
- \( r = 3 \) (number of successes)
The probability is calculated as follows:
Therefore, the probability of observing exactly 4 failures before 3 successes is approximately 0.1172, or 11.72%.
Further Reading
Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing more fun and curiosity to the world of science and research online.