The negative binomial distribution models how many independent yes-or-no trials are needed to get a fixed number of successes. It is useful when each trial has the same probability of success, such as repeated free throws, quality checks, or attempts to transmit a message. Unlike the binomial distribution, which fixes the number of trials and counts successes, the negative binomial fixes the number of successes and counts how long it takes.
This makes it a natural tool for waiting-time problems in probability and statistics.
Each trial is a Bernoulli trial with success probability p and failure probability 1 - p. If X is the total number of trials needed to get r successes, then the last trial must be a success, and the previous X - 1 trials must contain exactly r - 1 successes. This structure explains why combinations appear in the probability formula.
The geometric distribution is the special case of the negative binomial distribution when r = 1.
Key Facts
- If X = total trials needed to get r successes, then P(X = x) = C(x - 1, r - 1) p^r (1 - p)^(x - r), for x = r, r + 1, r + 2, ...
- The mean number of trials is E(X) = r / p.
- The variance of the number of trials is Var(X) = r(1 - p) / p^2.
- If Y = number of failures before the rth success, then P(Y = y) = C(y + r - 1, r - 1) p^r (1 - p)^y, for y = 0, 1, 2, ...
- The geometric distribution is the negative binomial distribution with r = 1.
- The binomial distribution fixes the number of trials n, while the negative binomial distribution fixes the number of successes r.
Vocabulary
- Bernoulli trial
- A Bernoulli trial is a random experiment with exactly two outcomes, usually called success and failure.
- Success probability
- The success probability p is the chance that one Bernoulli trial results in success.
- Negative binomial distribution
- The negative binomial distribution gives probabilities for the number of trials or failures needed to reach a fixed number of successes.
- Geometric distribution
- The geometric distribution gives probabilities for the number of trials needed to get the first success.
- Combination
- A combination C(n, k) counts how many ways k items can be chosen from n items when order does not matter.
Common Mistakes to Avoid
- Using the binomial formula when the number of trials is not fixed. This is wrong because negative binomial problems stop when a target number of successes is reached.
- Forgetting that the final trial must be a success. The factor C(x - 1, r - 1) is used because only the first x - 1 trials are arranged, while trial x is fixed as the rth success.
- Mixing up total trials and failures before success. If X counts total trials and Y counts failures, then X = Y + r, so the formulas use different exponents and domains.
- Changing the success probability from trial to trial without noticing. The standard negative binomial model requires independent trials with the same value of p each time.
Practice Questions
- 1 A basketball player makes a free throw with probability 0.70. What is the probability that the player's 5th made free throw occurs on the 8th attempt?
- 2 A machine produces a good part with probability 0.92. What is the expected number of parts that must be inspected to find 10 good parts?
- 3 A student says a negative binomial distribution should be used whenever a problem contains repeated trials and successes. Explain what extra conditions must be true before that model is appropriate.