A-LevelMathematicsProbability and statistics9709/61 · May/June 2024 · Q58 Marks

Sales of cell phones at a certain shop occur singly, randomly and independently.

(a)State one further condition that must be satisfied for the number of sales in a certain time period to be well modelled by a Poisson distribution.[1]
(b)The average number of sales per hour is 1.2 . Assume now that a Poisson distribution is a suitable model. Find the probability that the number of sales during a randomly chosen 12-hour period will be more than 12 and less than 16.[3]
(c)Use a suitable approximating distribution to find the probability that the number of sales during a randomly chosen 1-month period (140 hours) will be less than 150.[4]

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

[B1] Constant average rate
[B1] λ = 14.4
[M1] e^-14.4 (14.4^13/13! + 14.4^14/14! + 14.4^15/15!) or e^-14.4 (183837 + 189089 + 181526) or (0.102469 + 0.105396 + 0.101181)
[A1] = 0.309 (3sf)
[B1] N(140×1.2, 140×1.2) or N(168, 168)
[M1] (149.5 - 168) / √168 [= -1.427]
[M1] Ф(“-1.427") = 1 – Ф(“1.427”)
[A1] = 0.0768 or 0.0767 (3sf)

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