Negative Binomial Distribution Calculator
Calculate probabilities for the number of trials needed to achieve a fixed number of successes
Calculate Negative Binomial Distribution
Total number of trials (must be ≥ r)
Fixed number of successes needed
Probability of success on each trial (0 < p < 1)
Negative Binomial Distribution Results
Formula: P(Y = n) = C(n-1, r-1) × p^r × (1-p)^(n-r)
Combinations C(24, 14): 19,61,256
Mean Formula: μ = r/p = 15/0.4 = 37.500
Variance Formula: σ² = r(1-p)/p² = 15×0.600/0.4² = 56.250
Result Interpretation
Example: Leaflet Distribution
Problem Setup
Scenario: Handing out 15 leaflets on a street
Success probability: p = 0.4 (40% of people take a leaflet)
Question: What's the probability of handing out all leaflets in exactly 25 attempts?
Parameters: n = 25 trials, r = 15 successes, p = 0.4
Solution
P(Y = 25) = C(24, 14) × 0.4¹⁵ × 0.6¹⁰
C(24, 14) = 1,961,256
P(Y = 25) = 1,961,256 × 0.4¹⁵ × 0.6¹⁰ = 0.01273
This means there's about a 1.27% chance of distributing all leaflets in exactly 25 attempts.
Distribution Properties
Fixed Successes
Number of successes is predetermined
Random Variable
Number of trials needed (Y ≥ r)
Success Probability
Constant for each trial (0 < p < 1)
Examples
Rolling a die until getting three 6s
Trick-or-treating until collecting 20 candy bars
Flipping a coin until getting four heads
Soccer attempts until scoring three goals
Survey responses until getting desired sample size
Understanding the Negative Binomial Distribution
What is the Negative Binomial Distribution?
The negative binomial distribution models the number of trials needed to achieve a fixed number of successes in a sequence of independent Bernoulli trials. Unlike the binomial distribution, which fixes the number of trials and counts successes, the negative binomial fixes the number of successes and counts trials.
Key Characteristics
- •Fixed number of successes (r)
- •Variable number of trials (Y)
- •Constant success probability (p)
- •Independent trials
Mathematical Formulas
PMF: P(Y = n) = C(n-1, r-1) × p^r × (1-p)^(n-r)
Mean: μ = r/p
Variance: σ² = r(1-p)/p²
Parameter Definitions
- n: Number of trials
- r: Number of successes (fixed)
- p: Probability of success on each trial
- Y: Random variable (number of trials needed)
Comparison with Binomial Distribution
| Aspect | Binomial | Negative Binomial |
|---|---|---|
| Fixed Parameter | Number of trials (n) | Number of successes (r) |
| Random Variable | Number of successes (X) | Number of trials (Y) |
| Range | 0 ≤ X ≤ n | Y ≥ r |
| Question Type | How many successes in n trials? | How many trials for r successes? |