A-LevelMathematicsProbability and statistics9709/62 · Oct/Nov 2020 · Q59 Marks
Customers arrive at a shop at a constant average rate of 2.3 per minute. It is now given that the number of customers arriving per minute has the distribution Po(2.3).
- (a)State another condition for the number of customers arriving per minute to have a Poisson distribution.[1]
- (b)Find the probability that exactly 3 customers arrive during a 1-minute period.[2]
- (c)Find the probability that more than 3 customers arrive during a 2-minute period.[3]
- (d)Five 1-minute periods are chosen at random. Find the probability that no customers arrive during exactly 2 of these 5 periods.[3]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.
- [B1] Customers arrive independently or singly or at random (Any one of these in context)
- [M1] e⁻²·³ × 2.3³ / 3!
- [A1] = 0.203 (3sf)
- [B1] Po(4.6) SOI
- [M1] 1 - e⁻⁴·⁶ (1 + 4.6 + 4.6²/2! + 4.6³/3!) (Correct expression, with any λ, allow one end error)
- [A1] = 0.674 (3sf) As final answer.
- [M1] Calculate P(no customers arrive in 1 minute) = e⁻²·³ (= 0.10026)
- [M1] Apply Binomial probability: ⁵C₂ (e⁻²·³)² (1 - e⁻²·³)^3
- [A1] = 0.0732 or 0.0733 (3sf)
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