Geometric and Negative Binomial Distributions: Exam Tips

The geometric distribution answers a simple waiting question: how many trials will it take to get the first success? If each basketball free throw goes in with probability 0.7, how many throws until the first basket? The related negative binomial distribution extends this to the r-th success: how many trials until the second, third or r-th success?

These matter because real life is full of waiting: how many customers walk in before the first one buys, how many emails before the first reply, how many machine runs before the second failure. The binomial counts successes in a fixed number of trials; these two distributions flip the question and fix the number of successes instead.

Key Concepts

• p: success probability per trial, constant, with independent trials, just like the binomial.
• Geometric X: the trial number of the FIRST success. P(X = k) = q^(k-1) × p, for k = 1, 2, 3, …, with mean 1/p and variance q/p².
• Negative binomial Y: the trial number of the r-TH success. P(Y = k) = C(k-1, r-1) × p^r × q^(k-r), for k = r, r+1, …, with mean r/p and variance rq/p².
• Memoryless property (geometric only): past failures do not change the distribution of the remaining wait; the past is forgotten.
• Two conventions: some books count failures before the first success (values 0, 1, 2, …) instead of the trial of success. Know which your exam uses.

Formulas

Geometric: P(X = k) = (1-p)^(k-1) × p, mean 1/p, variance (1-p)/p². Negative binomial: P(Y = k) = C(k-1, r-1) × p^r × (1-p)^(k-r), mean r/p, variance r(1-p)/p². Both assume independent trials with constant p, and the negative binomial needs at least r trials. When r = 1, it collapses exactly to the geometric.

Worked Example

A quiz app shows questions until you answer one correctly; each question is answered correctly with probability p = 0.25. (a) Probability the first correct answer is on the 3rd question: q²p = (0.75)² × 0.25 = 0.5625 × 0.25 = 0.140625. The mean wait is 1/0.25 = 4 questions. (b) Probability the 2nd correct answer comes on the 4th question: C(3, 1) × 0.25² × 0.75² = 3 × 0.0625 × 0.5625 = 3 × 0.035156 = 0.105469. The mean for the 2nd success is 2/0.25 = 8 questions.

Common Misconceptions

• Negative binomial is not "binomial with a negative": the name is historical. It fixes the number of successes and counts trials; the binomial fixes trials and counts successes.
• Do not start counting at 0 by accident: with the trial-of-success convention, the geometric support begins at k = 1, never 0. If your book counts failures first, the mean is q/p, not 1/p.
• The last trial must be a success: in the negative binomial, the k-th trial is always the r-th success, which is why the coefficient is C(k-1, r-1) and not C(k, r).

Real-World Applications

• Sales: how many cold calls before the first sale, and how many before the fifth sale for weekly targets.
• Reliability: how many production runs before the second machine fault, to schedule maintenance.
• Medicine: how many patients must be screened until the third positive case in a trial.

Geometric and negative binomial are the waiting-time partners of the binomial: the first uses q^(k-1)p with mean 1/p for the first success, the second uses C(k-1, r-1)p^r q^(k-r) with mean r/p for the r-th success.