A-LevelMathematicsProbability and statistics9709/62 · Feb/Mar 2023 · Q213 Marks

The number of orders arriving at a shop during an 8-hour working day is modelled by the random variable X with distribution Po(25.2).

(a)State two assumptions that are required for the Poisson model to be valid in this context.[2]
(b)(i)Find the probability that the number of orders that arrive in a randomly chosen 3-hour period is between 3 and 5 inclusive.[3]
(b)(ii)Find the probability that, in two randomly chosen 1-hour periods, exactly 1 order will arrive in one of the 1-hour periods, and at least 2 orders will arrive in the other 1-hour period.[4]
(c)The shop can only deal with a maximum of 120 orders during any 36-hour period. Use a suitable approximating distribution to find the probability that, in a randomly chosen 36-hour period, there will be too many orders for the shop to deal with.[4]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2023). The mark or mark type is shown in brackets.

[B1] Orders arrive at constant mean rate (must say mean or rate)
[B1] Orders arrive at random
[B1] Orders arrive independently
[B1] Orders arrive singly
[B1] Must be in context (accept 25.2 as context).
[B1] Any one reason correctly stated.
[B1] A second reason correctly stated.
[SC B1] both correct, not in context.
[B1] λ = 3/8 × 25.2 [= 9.45]
[M1] e^-9.45 * (9.45^3/3! + 9.45^4/4! + 9.45^5/5!) or e^-9.45 * (140.65 + 332.29 + 628.03) or 0.01107 + 0.02615 + 0.04942
[M1] Allow any λ. Allow end errors. Expression must be seen.
[A1] = 0.0866 (3 sf)
[A1] If M0 allow SC B1 for 0.0866 no working seen.
[B1] e^-3.15 * 3.15 or (1 – e^-3.15(1 + 3.15)) or 0.135 or 0.822 (3 sf)
[M1] M1 for product of two Poisson probabilities P(1) × (1–P(0,1)) (no end errors accepted). Accept any λ.
[M1] M1 for their product of two Poisson probabilities (accept end errors) × 2. Accept any λ
[A1] 0.222 (3 sf)
[B1] N(113.4, 113.4) SOI
[M1] (120.5-113.4) / sqrt(113.4) [= 0.667]
[M1] Standardise with their values. Allow wrong or no cc. Must have sqrt.
[M1] 1 – Φ(their ‘0.667')
[M1] For probability area consistent with their values.
[A1] = 0.252 (3 sf)

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