The scaling step that costs most of the marks, and the identity that tells you the model is wrong.
Stretch the window to two hours and ask for λ.
Many will say 3, because 3 is the number in the question. It is 6, and the readout says so while the quoted rate has not changed at all.
Note what the mean and variance readouts do: they move together, always, because they are the same number. That is the identity examiners use when they ask whether Poisson is appropriate, and it arrives here for free.
A binomial has a fixed n and stops there. A Poisson has no n and no largest value. The language in the question is the tell: “out of 50” is binomial, “per hour” or “per page” is Poisson.
Give them five one-line scenarios and have them sort rather than compute. It takes four minutes and it prevents the wrong distribution being chosen under exam pressure, which is unrecoverable.
| 1. λ for 250 m | 2 × (250/100) = 5. |
| 2. sd of Po(9) | Variance is λ = 9, so sd = 3. |
| 3. P(X = 0) for Po(1) | e−1 = 0.368. |
| 4. Mean 4.1, variance 11.8 | B, Poisson fits poorly. The variance is nearly three times the mean, so the events are clumping. |
1 markScaling λ to the interval the question asks about, before any calculator work.
1 markReading the inequality correctly, with “at least 2” as 1 − P(X ≤ 1).
1 markJustifying the model: constant average rate, independent events, mean roughly equal to variance.
| They give | What it means |
|---|---|
| 2 (Q1) | Used the quoted rate unchanged. The single most common Poisson error and the reason for the widget. |
| 500 (Q1) | Multiplied by 250 instead of by 250/100. |
| 2.5 (Q1) | Found the number of 100 m lengths and forgot to multiply by the rate. |
| 9 (Q2) | Gave the variance. It equals λ, which is the trap: the number looks like an answer. |
| 0.632 (Q3) | 1 minus the answer, so they found P(X ≥ 1). |
| 0 (Q3) | Thought zero arrivals impossible. It is the single most likely count when λ = 1. |
| A or C (Q4) | Either missed that the test is mean against variance, or fitted a model they had just disproved. Both are worth naming out loud. |
"Can X be bigger than λ?" Far bigger. There is no upper limit at all, which is the structural difference from the binomial and worth repeating.
"What if the rate changes during the day?" Then Poisson is the wrong model for the whole day, and noticing that is the mark on a modelling question. A single peak hour can still be Poisson on its own.
"Why e?" It falls out of the limit of a binomial with n large and p small. Worth one sentence, not a derivation, unless they ask twice.
| What is happening | |
|---|---|
| 1 | Stretch the window. Ask for λ at two hours. Let the wrong answer be said out loud first. |
| 2 | Sorting five scenarios into Poisson or binomial. |
| 3 | The formula, the calculator, and the four questions. |
| 4 | A question needing two intervals added, which is where λ adding becomes useful rather than abstract. |
| 5 | The mean-against-variance check on a real data set, with a written verdict. |
Do not write λ = 3 on the board and leave it there for the whole lesson. It is the thing that gets copied into a two-hour question.
Do not present the mean-equals-variance identity as a curiosity. It is the diagnostic, and questions award marks for applying it.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.