Maths › Further Statistics 1 › Geometric and negative binomial distributions
Geometric and negative binomial distributions
Stop counting successes in a fixed number of trials and start counting trials until a fixed number of successes: the question turns round, and so does the distribution.
Builds on The Poisson distribution and Geometric series.
IN THIS TOPIC
- Model the trial of a first success with the geometric distribution.
- Extend to the rth success with the negative binomial distribution.
- Quote and apply the means and variances of both.
WHAT YOU PROBABLY THINK
Waiting for a success is just the binomial distribution read backwards, so the same formulae apply.
Waiting for the first success
The geometric distribution counts the trial on which the first success arrives: x − 1 failures then a success, so P(X = x) = p(1 − p)x-1. The binomial fixes the trials and counts successes; the geometric fixes the successes and counts trials, and the opening claim misses that the roles have swapped entirely. There is no upper limit to X, and the probabilities decay geometrically, which is where the name comes from.
WORKED EXAMPLE
A first success
A spinner lands on red with probability 0.2. Find the probability the first red is on the third spin, and the mean number of spins needed.
P(X = 3) = 0.2 × 0.8² = 0.128.
μ = 1/p = 5 spins, with variance 0.8/0.04 = 20.
The mode is always 1, however small p is: the single likeliest trial for a first success is the first one, even when the average wait is long.
Waiting for the rth success
The negative binomial generalises it: for the rth success to land on trial x, the first x − 1 trials must hold exactly r − 1 successes, and trial x must succeed:
The combination counts where the earlier successes sit, and the final trial is pinned rather than chosen. Setting r = 1 recovers the geometric distribution exactly, means and variances included.
WORKED EXAMPLE
The third success
Trials succeed independently with probability 0.4. Find the probability the third success occurs on the fifth trial, and the mean number of trials for three successes.
P(X = 5) = ⁴C₂ × 0.4³ × 0.6² = 6 × 0.064 × 0.36 = 0.138.
μ = r/p = 3/0.4 = 7.5 trials, with variance 3 × 0.6/0.16 = 11.25.
The mean is three times the geometric mean, as it must be: three independent waits, one after another.
YOUR TURN
Recognising the model
A quality inspector tests items until finding the second faulty one. Faults occur independently with probability 0.1. Find the probability this happens on the fifth item tested.
Show the working
This is negative binomial with r = 2 and p = 0.1.
P(X = 5) = ⁴C₁ × 0.1² × 0.9³ = 4 × 0.01 × 0.729.
= 0.0292. The combination counts the four places the first fault could have taken among the earlier trials.
THE EXAM BIT
- Read what is being counted: trials until a success is geometric or negative binomial, successes in n trials is binomial.
- The last trial is always a success in these models, so the combination only arranges the earlier ones.
- For 'at least x trials', use (1 − p) to the power x − 1: no summation is needed.
- Quote the means and variances rather than deriving them unless the question asks for a derivation.
CHECK YOURSELF
A biased coin shows heads with probability 0.25. Find the probability that the first head appears on the fourth toss, and the mean number of tosses required.
Show a hint
Geometric with p = 0.25.
Show the answer
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Geometric: P(X = x) = p(1 − p)x-1, mean 1/p, variance (1 − p)/p².
Negative binomial for the rth success: the combination arranges the earlier successes, mean r/p.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Model the trial of a first success with the geometric distribution.
- Extend to the rth success with the negative binomial distribution.
- Quote and apply the means and variances of both.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.