Poisson model and probabilitiesAQA A-Level Further Maths: Subtopic test

1. 1
   Faults occur at random, independently and at a constant average rate of 2.5 per metre in a long roll of fabric. The number of faults, $X$, in one metre is modelled by $X\sim\mathrm{Po}(2.5)$.

   (a)

   Find $\mathrm{P}(X=2)$.

   [1 mark]

   • A$0.2052$
   • B$0.2565$
   • C$0.5438$
   • D$0.2138$

   (b)

   Find the probability that there is at least one fault in a metre of fabric.

   [1 mark]

   • A$0.9179$
   • B$0.0821$
   • C$0.7127$
   • D$0.2873$

   (c)

   Find the probability that a 2 metre length of the fabric contains no faults.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A call centre receives calls at random at a constant average rate of 6 per hour. The number of calls, $Y$, received in one hour is modelled by $Y\sim\mathrm{Po}(6)$.

   (a)

   Find the standard deviation of $Y$.

   [1 mark]

   • A$6$
   • B$36$
   • C$2.45$
   • D$3$

   (b)

   Which of the following is a necessary condition for the Poisson model to be valid for the calls?

   [1 mark]

   • ACalls arrive at exactly equal intervals
   • BA call is more likely just after a busy hour
   • CThe average number of calls differs from one hour to the next
   • DCalls occur independently of one another

   (c)

   The manager notices that calls are much more frequent between 12 noon and 1 pm than at other times of the day. Explain why $\mathrm{Po}(6)$ is not suitable as a model for the number of calls in every hour of the working day.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The number of typing errors on a page of a manuscript is modelled by $X\sim\mathrm{Po}(1.8)$. Errors on different pages occur independently of one another.

   (a)

   Find $\mathrm{P}(X\ge3)$.

   [3 marks]

   (b)

   Five pages are chosen at random. Find the probability that exactly two of them contain no errors.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The number of accidents at a road junction in a month is modelled by $X\sim\mathrm{Po}(\lambda)$. Records show that in 20% of months there are no accidents at the junction.

   (a)

   (i) Show that $\lambda=\ln5$ and write down its value to 3 significant figures.
   (ii) State two conditions needed for a Poisson model to be valid in this context.
   (iii) Find $\mathrm{P}(X=2)$.

   [6 marks]

   (b)

   (i) Find the probability that the number of accidents in a month is greater than its mean.
   (ii) Find the probability that the number of accidents in a month is within one standard deviation of the mean.

   [6 marks]

   Total for question 4: 12 marks

End of questions