Understanding the Negative Binomial Distribution
Negative Binomial PMF (failures before the r-th success):
P(X = k) = C(k + r - 1, k) × pr × (1 - p)k
r = number of successes to reach; k = number of failures before the r-th success (k = 0, 1, 2, …); p = probability of success on each trial. C(k+r-1, k) is the binomial coefficient "k+r-1 choose k".
The negative binomial distribution answers a question the ordinary binomial cannot: instead of fixing the number of trials and counting successes, it fixes the number of successes r and asks how many failures k pile up before you get there. Each trial is an independent Bernoulli experiment with the same success probability p. The last trial in the sequence is always a success (the r-th one), which is why the coefficient is C(k+r-1, k) rather than C(k+r, k) — you are only free to arrange the first k failures and r−1 successes among the first k+r−1 trials.
Why the formula has each piece
- pr: every valid sequence contains exactly r successes, each contributing a factor of p.
- (1 - p)k: it also contains exactly k failures, each contributing a factor of (1 - p).
- C(k + r - 1, k): the number of distinct orderings of those k failures among the k+r−1 trials that precede the final, guaranteed r-th success.
Mean, variance, and over-dispersion
For this "number of failures" parameterization the expected number of failures is μ = r(1 - p)/p and the variance is σ² = r(1 - p)/p². Notice that the variance is always the mean divided by p, so it is strictly larger than the mean whenever p < 1. This "variance > mean" property, called over-dispersion, is exactly why statisticians reach for the negative binomial to model count data (insurance claims, disease cases, website purchases) that is too spread out for a Poisson model, where variance and mean are forced to be equal.
A worked example
Suppose you flip a fair coin (p = 0.5) and want the 3rd head (r = 3). What is the probability of exactly 2 tails before that 3rd head (k = 2)? Then C(2+3−1, 2) = C(4, 2) = 6, so P(X = 2) = 6 × 0.5³ × 0.5² = 6 × 0.03125 = 0.1875. On average you would expect r(1−p)/p = 3(0.5)/0.5 = 3 tails before the 3rd head.
Frequently Asked Questions
What does the negative binomial distribution model?
How is it different from the binomial distribution?
Why does k start at 0 instead of 1?
Can p be exactly 1?
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|---|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
| Hotel housekeeping | $2–5/night | Leave daily, not just at checkout |
| Barber / hair salon | 15–20% | On pre-discount total |
| Pizza delivery | 10–15% or $3–5 minimum | Higher in bad weather |
General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|---|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|---|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |