Counting random events in a fixed window: Picture a quiet help-desk that gets, on average, 3 calls per hour. Some hours bring 1 call, some bring 6, some bring none — the calls arrive at random.
The Poisson distribution gives the probability of getting exactly x events in that window when events happen independently at a steady average rate.
It has just one parameter, the mean m (the average number of events). Write X ~ Po(m).
Why this shape?: The mˣ grows with more events, 1/x! punishes large counts (x! explodes), and e−m is the normalising constant that makes all the probabilities sum to 1.
x can be any non-negative whole number — there is no upper limit, unlike the binomial.
IB-style question — exactly x events by formula
A help-desk receives calls at an average rate of 3 per hour, modelled by X ~ Po(3).
Find the probability of exactly 2 calls in a given hour.
Step by step
- Write the formula with m = 3 and x = 2.
- Substitute: 3² = 9 and 2! = 2.
- Evaluate.
Final answer
P(X = 2) = 0.224 (3 s.f.).
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Let the GDC do the arithmetic: A GDC is allowed on every AI paper, so you rarely plug into the formula by hand.
• poissonpdf(m, x) gives P(X = x) (exactly x). • poissoncdf(m, x) gives P(X ≤ x) (up to and including x).
For ranges, combine these: P(X ≥ k) = 1 − P(X ≤ k − 1) and P(X > k) = 1 − P(X ≤ k).
The Poisson's special property: mean = variance: For X ~ Po(m), the mean is m and the variance is also m — so the standard deviation is √m.
This is unique to the Poisson and is a quick suitability check: if a data set has its sample mean ≈ its sample variance, a Poisson model is plausible.
IB-style question — 'at least' on the GDC
Cars pass a remote junction at an average of 4.5 per minute, modelled by X ~ Po(4.5).
Find the probability that at least 6 cars pass in a given minute.
Step by step
- 'At least 6' is the complement of 'at most 5'.
- On the GDC: P(X ≤ 5) = poissoncdf(4.5, 5).
- Subtract from 1.
Final answer
P(X ≥ 6) = 0.297 (3 s.f.) — about a 30% chance of a busy minute.
Independent Poissons just add: If two independent Poisson counts have means m₁ and m₂, their total is also Poisson with mean m₁ + m₂.
This lets you rescale the window: a rate of 3 calls per hour over 4 hours is X ~ Po(3 × 4) = Po(12). Multiply the rate by the length of the interval.
IB-style question — combine two sources
A wildlife camera films two ponds. Pond A is visited by birds at a mean of 2.1 per hour and pond B at 1.4 per hour, independently, each modelled by a Poisson distribution.
Find the probability that exactly 3 birds in total are filmed in one hour.
Step by step
- Independent Poissons add: the total T has mean 2.1 + 1.4.
- Want exactly 3 — use poissonpdf(3.5, 3).
- Evaluate on the GDC.
Final answer
P(T = 3) = 0.216 (3 s.f.).
IB-style question — Poisson then binomial
A field is split into sections. The number of worms per section is Poisson with mean 1.5.
Find the probability that exactly 5 of 6 randomly chosen sections each contain at least one worm.
Step by step
- First, the chance a single section has at least one worm.
- 'At least one worm' is now a success with p = 0.7769 over n = 6 independent sections — a binomial model.
- Find P(Y = 5) on the GDC (binompdf).
Final answer
P(Y = 5) ≈ 0.379 (3 s.f.). The key move: a Poisson probability becomes the success probability p of a binomial across several trials.