The Poisson distributionEdexcel International A Level Further Maths: Subtopic test

1. 1
   Typing errors occur at random in a manuscript at a mean rate of 1.5 per page. The number of errors on one page is modelled by $X\sim\text{Po}(1.5)$.

   (a)

   Find $P(X=2)$.

   [1 mark]

   • A$0.251$
   • B$0.809$
   • C$0.223$
   • D$0.558$

   (b)

   Errors on different pages are independent. Find the variance of the number of errors in a 4-page section.

   [1 mark]

   • A$1.5$
   • B$2.45$
   • C$36$
   • D$6$

   (c)

   Find the probability that a 2-page section contains no errors.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A call centre receives calls at random at a constant mean rate of 4 calls per 10 minutes.

   (a)

   Which distribution models the number of calls received in a 25-minute period?

   [1 mark]

   • A$\text{Po}(4)$
   • B$\text{Po}(10)$
   • C$\text{Po}(1.6)$
   • D$\text{Po}(100)$

   (b)

   Let $X$ be the number of calls in a 10-minute period. Find $P(X<4)$.

   [1 mark]

   • A$0.238$
   • B$0.195$
   • C$0.433$
   • D$0.629$

   (c)

   Find the probability that more than 6 calls are received in a 10-minute period.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Flaws occur at random in a roll of fabric at a mean rate of 0.4 per square metre. Flaws in separate pieces of fabric occur independently.

   (a)

   Find the probability that a 5 m$^2$ piece of fabric contains at most 3 flaws.

   [3 marks]

   (b)

   A customer buys one 5 m$^2$ piece and one separate 2.5 m$^2$ piece. Find the probability that together they contain more than 4 flaws.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A bus company states that buses arrive at a stop at random at a mean rate of 6 per hour during the morning, and that the number of buses arriving in any interval can be modelled by a Poisson distribution.

   (a)

   (i) State two conditions under which a Poisson distribution is a suitable model for the number of buses arriving.
   (ii) Suggest one reason why a Poisson distribution may not be suitable for buses.
   (iii) Using the model, find the probability that at least 3 buses arrive in a 20-minute period.

   [6 marks]

   (b)

   Assume the model is valid.
   (i) Find the probability that no bus arrives in a 10-minute period.
   (ii) Find the probability that exactly 2 buses arrive in the first 15 minutes of an hour and exactly 3 buses arrive in the next 15 minutes.
   (iii) Explain why the answer to (ii) is not the probability of exactly 5 buses arriving in the first 30 minutes.

   [6 marks]

   Total for question 4: 12 marks

End of questions