Further Statistics 1: Poisson and binomial distributionsEdexcel A-Level Further Maths: Topic test

1. 1
   Potholes along a country road occur randomly and independently, at a mean rate of $1.5$ per kilometre.

   (a)

   Which distribution models the number of potholes in a $2$ km stretch of the road?

   [1 mark]

   • A$\mathrm{Po}(1.5)$
   • B$\mathrm{Po}(3)$
   • C$\mathrm{Po}(0.75)$
   • D$\mathrm{Po}(3.5)$

   (b)

   Let $X$ be the number of potholes in a $2$ km stretch. What is $P(X=2)$?

   [1 mark]

   • A$0.2510$
   • B$0.4232$
   • C$0.0498$
   • D$0.2240$

   (c)

   Find the probability that a randomly chosen $1$ km stretch contains at least one pothole.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Kingfishers and herons are seen on a stretch of river at random and independently. The number of kingfishers seen per hour has a Poisson distribution with mean $0.6$, and the number of herons seen per hour has a Poisson distribution with mean $1.4$.

   (a)

   Which distribution models the total number of these birds seen in one hour?

   [1 mark]

   • A$\mathrm{Po}(2.0)$
   • B$\mathrm{Po}(0.84)$
   • C$\mathrm{Po}(1.0)$
   • D$\mathrm{Po}(0.8)$

   (b)

   What is the probability that, in a three-hour period, at most one bird in total is seen?

   [1 mark]

   • A$0.0025$
   • B$0.0149$
   • C$0.0174$
   • D$0.0620$

   (c)

   Find the probability that exactly one kingfisher is seen in a two-hour period.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The random variable $Y\sim\mathrm{B}(n,p)$ has mean $4.5$ and variance $3.15$.

   (a)

   Find the values of $n$ and $p$.

   [3 marks]

   (b)

   Find $P(3\le Y\le6)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A seed supplier sells packets of $250$ seeds. Each seed independently fails to germinate with probability $0.008$. Let $X$ be the number of seeds in a packet that fail to germinate.

   (a)

   (i) Give two reasons why a Poisson distribution is suitable to approximate the distribution of $X$.
   (ii) Use a Poisson approximation to find $P(X=2)$.
   (iii) Find $P(X=2)$ using the binomial distribution and comment on your two answers.

   [6 marks]

   (b)

   A customer buys three packets. Using the Poisson approximation for each packet, let $T$ be the total number of seeds in the three packets that fail to germinate.
   (i) State the distribution of $T$.
   (ii) Find $P(T\ge8)$.
   (iii) The customer buys three packets on each of two days. Assuming independence, find the probability that $T\ge8$ on exactly one of the two days.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Meteors are observed at random, independently of one another, at a mean rate of $3.6$ per hour.

   (a)

   Which of these is an assumption needed to model the number of meteors in a fixed time with a Poisson distribution?

   [1 mark]

   • AMeteors occur at regular intervals.
   • BThe mean rate changes through the night.
   • CMeteors occur independently of one another.
   • DNo more than $3$ meteors can occur in any hour.

   (b)

   What is the probability that at least $2$ meteors are observed in a $30$ minute period?

   [1 mark]

   • A$0.537$
   • B$0.463$
   • C$0.297$
   • D$0.874$

   (c)

   Find the probability that more than $3$ meteors are observed in a one hour period.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The random variable $X\sim\mathrm{B}(40,0.35)$.

   (a)

   What is $\mathrm{Var}(X)$?

   [1 mark]

   • A$14$
   • B$0.2275$
   • C$26$
   • D$9.1$

   (b)

   A student suggests approximating $X$ by $\mathrm{Po}(14)$. Which statement is the best reason why this is unsuitable?

   [1 mark]

   • A$n$ is too large for a Poisson approximation.
   • B$p$ is not small, and the variance of $X$ is $9.1$ whereas a Poisson variable with mean $14$ has variance $14$.
   • CThe mean of $X$ is not $14$.
   • DThe probability of success must be exactly $0.5$.

   (c)

   Find $P(X=12)$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The number of scratches, $X$, on a metal panel has a Poisson distribution with mean $\lambda$. Exactly $20\%$ of panels have no scratches.

   (a)

   Find the value of $\lambda$.

   [3 marks]

   (b)

   (i) Find the probability that a panel has at least $3$ scratches.
   (ii) A sample of $150$ panels has a mean of $1.58$ scratches and a variance of $2.9$. Comment on the suitability of the Poisson model.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Sachets of seed are packed in boxes of $150$. Each sachet is independently underweight with probability $0.02$. Let $X$ be the number of underweight sachets in a box.

   (a)

   (i) Find $\mathrm{E}(X)$ and $\mathrm{Var}(X)$.
   (ii) Use a Poisson approximation to find the probability that a box has at least $5$ underweight sachets.
   (iii) A pallet holds $12$ boxes. Using your answer to part (ii), find the probability that exactly $2$ boxes on the pallet have at least $5$ underweight sachets.

   [6 marks]

   (b)

   A second machine $Q$ packs boxes in which the number of underweight sachets has a Poisson distribution with mean $1.5$, independent of machine $P$. A retailer opens one box packed by the first machine, with $X$ approximated by $\mathrm{Po}(3)$, and one box packed by $Q$. Let $U$ be the total number of underweight sachets in the two boxes.
   (i) Explain why $U\sim\mathrm{Po}(4.5)$.
   (ii) Find $P(U=4)$.
   (iii) Find $P(U\ge8)$.

   [6 marks]

   Total for question 8: 12 marks

End of questions