The Poisson distributionEdexcel A-Level Further Maths: Subtopic test

1. 1
   The number of emails, $X$, received by an office in a 10-minute period is modelled by $X\sim\text{Po}(3.5)$.

   (a)

   Find $P(X=2)$.

   [1 mark]

   • A$0.185$
   • B$0.321$
   • C$0.106$
   • D$0.0302$

   (b)

   Find $P(X\geq3)$.

   [1 mark]

   • A$0.321$
   • B$0.463$
   • C$0.679$
   • D$0.185$

   (c)

   Find $P(2<X\leq6)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A helpline receives calls at random, independently of each other, at a mean rate of 6 per hour.

   (a)

   Which distribution models the number of calls received in a 20-minute period?

   [1 mark]

   • A$\text{Po}(6)$
   • B$\text{Po}(20)$
   • C$\text{Po}(12)$
   • D$\text{Po}(2)$

   (b)

   Find the probability that no calls are received in a 20-minute period.

   [1 mark]

   • A$0.271$
   • B$0.135$
   • C$0.865$
   • D$0.00248$

   (c)

   Find the probability that exactly one call is received in each of two successive 20-minute periods.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A café receives online orders at a mean rate of 3 per hour and phone orders at a mean rate of 2 per hour. Both types of order arrive at random and independently of each other.

   (a)

   Find the probability that at least 2 orders in total are received in a 30-minute period.

   [3 marks]

   (b)

   A 15-minute period is called busy if at least 4 orders are received in it. Find the probability that exactly 2 of 3 successive 15-minute periods are busy.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A traffic engineer records the vehicles passing a point on a country road between 10 am and noon. The mean number of vehicles per minute is $3.1$. The engineer models the number of vehicles passing in one minute by a Poisson distribution. Vehicles on this road often travel in convoys behind slow lorries.

   (a)

   (i) Suggest a suitable value of $\lambda$ and state two conditions required for a Poisson model.
   (ii) Using this model, find the probability that more than 5 vehicles pass in one minute.
   (iii) Explain why the Poisson model may not be appropriate for this road.

   [6 marks]

   (b)

   (i) Find the probability that at most 10 vehicles pass in a 5-minute period.
   (ii) A second country road has vehicles passing at a mean rate of $2.4$ per minute, independently of the first road. Find the probability that exactly 5 vehicles in total pass in one minute on the two roads.

   [6 marks]

   Total for question 4: 12 marks

End of questions