When the binomial has nothing to count
Poisson distribution — The distribution of the number of events occurring in a fixed interval, when events happen independently at a constant average rate λ.
The binomial distribution needs a fixed number of trials. Ask how many phone calls a switchboard receives in an hour, or how many flaws appear in a metre of cable, and there is no n at all — there is no natural list of trials that either succeeded or failed.
What these situations share instead is a constant average rate of occurrence over a continuous interval of time, length or area. The Poisson distribution models exactly that.
- λ
- the mean number per intervalmust match the interval in the question
- r
- the countany non-negative integer, with no upper limit
- e^(−λ)
- the normalising factorwhat makes the probabilities total 1
At small λ the distribution is crushed against zero and heavily skewed. Push λ past about 10 and it becomes visibly bell-shaped — which is precisely why the normal approximation is allowed there and not before.
The conditions, and the interval trap
Three conditions must hold, and questions frequently ask you to state them. Events must occur independently, at a constant average rate, and singly — not in clusters arriving together.
The most common calculation error has nothing to do with the formula. λ must match the interval the question asks about. If calls arrive at 6 per hour and you are asked about a 20-minute period, then λ = 2, not 6. Scaling λ is the first line of the answer, not an afterthought.
| Given rate | Question asks about | λ to use |
|---|---|---|
| 5 per hour | one hour | 5 |
| 5 per hour | 30 minutes | 2.5 |
| 5 per hour | 3 hours | 15 |
| 2 per metre | 4.5 metres | 9 |
| 0.4 per page | 20 pages | 8 |
Scale λ before anything else
Write the scaled λ on its own line at the start. It costs a second, it is often worth a mark in itself, and it prevents the whole question being marked wrong for an otherwise perfect method. If the interval doubles, λ doubles — the rate is per unit, so it scales linearly.
Adding Poissons, and approximating a binomial
Two independent Poisson variables add in the simplest possible way: the sum is Poisson again, with the two means added. This is why a switchboard receiving 3 calls an hour on one line and 5 on another can be treated as a single Po(8).
The Poisson also rescues the binomial when n is large and p is small. Computing ⁵⁰⁰C₃(0.004)³(0.996)⁴⁹⁷ is unpleasant; approximating with Po(np) is not. The usual conditions are n > 50 and np < 5 — large n so events are spread thin, small p so they are rare.
- a + b
- the combined meanvalid only if the two are independent
- np
- the approximating meanthe binomial mean carried across
- N(λ, λ)
- the normal approximationmean and variance both λ
Faults occur in a cable at an average rate of 0.6 per metre. Find the probability that a 5-metre length contains (a) exactly 2 faults, (b) at least 2 faults.
- Scale the mean: λ = 0.6 × 5 = 3 for a 5-metre length.The rate is per metre and the interval is five metres, so λ multiplies by 5. This is the step that most often goes wrong.
- (a) P(X = 2) = e⁻³ · 3² / 2! = 0.049787 × 9 / 2.Direct substitution into the Poisson formula with r = 2.
- = 0.224.To 3 significant figures.
- (b) P(X ⩾ 2) = 1 − P(X = 0) − P(X = 1).The complement is far shorter than summing an infinite tail — and the tail really is infinite here, so it is the only practical route.
- P(X = 0) = e⁻³ = 0.0498; P(X = 1) = e⁻³ × 3 = 0.1494.The r = 0 term is just e^(−λ), since λ⁰ = 1 and 0! = 1.
- P(X ⩾ 2) = 1 − 0.0498 − 0.1494 = 0.801."At least 2" includes 2, so only 0 and 1 are removed.
(a) 0.224; (b) 0.801
Recognising which distribution to use
- A fixed number of trials, each success or failure → binomial.
- A rate over time, length or area with no fixed n → Poisson.
- Mean equal to variance in the data → a strong hint the model is Poisson.
- Large n, small p, and awkward arithmetic → approximate the binomial by Po(np).
- λ above about 15 → approximate the Poisson by N(λ, λ), with a continuity correction.
Recognising a Poisson in a worded question
Questions rarely say which distribution to use. They describe a situation, and the phrasing itself carries the answer if you know what to listen for.
The giveaway is that a Poisson situation has no natural denominator. "Three calls per hour" gives no number of trials — you cannot say how many calls did not happen. A binomial situation always has one: twenty components tested, fifty coin tosses, a fixed n.
A second clue is the interval. Poisson questions describe events spread over time, length, area or volume, and the interval can be rescaled at will. Binomial questions describe a fixed batch that cannot be resized.
| Wording | Distribution | Why |
|---|---|---|
| "4 flaws per metre of cable" | Poisson | a rate; no count of non-flaws |
| "of 50 components, 3% are faulty" | binomial | fixed n = 50 |
| "calls arrive at 6 per hour" | Poisson | a rate over time |
| "20 patients, each 10% likely" | binomial | fixed n = 20 |
| "accidents per month at a junction" | Poisson | a rate over time |
The mean-equals-variance check
If a question gives you sample data rather than a model, compare the mean with the variance. Roughly equal supports Poisson. A variance much larger than the mean means the events cluster rather than occurring independently, and a variance much smaller means they are more evenly spaced than chance would give — neither is Poisson, and saying so is often the mark.