Poisson approximation to the binomialEdexcel International A Level Further Maths: Subtopic test

1. 1
   A factory produces light bulbs. Each bulb is independently defective with probability 0.02. A box contains 200 bulbs, and $X$ is the number of defective bulbs in a box, so $X\sim B(200,0.02)$.

   (a)

   Which Poisson distribution is the best approximation to the distribution of $X$?

   [1 mark]

   • A$\text{Po}(0.02)$
   • B$\text{Po}(4)$
   • C$\text{Po}(200)$
   • D$\text{Po}(10000)$

   (b)

   Use a Poisson approximation to find $P(X\le2)$.

   [1 mark]

   • A$0.147$
   • B$0.092$
   • C$0.238$
   • D$0.433$

   (c)

   Use a Poisson approximation to find the probability that a box contains more than 5 defective bulbs.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   An airline finds that each passenger booked on a flight independently fails to turn up with probability 0.03. A flight has 120 passengers booked, and $X$ is the number who fail to turn up, so $X\sim B(120,0.03)$.

   (a)

   Find the mean of the Poisson distribution that approximates $X$.

   [1 mark]

   • A$3.6$
   • B$36$
   • C$0.03$
   • D$4000$

   (b)

   Use the Poisson approximation to find the probability that every passenger turns up, $P(X=0)$.

   [1 mark]

   • A$0.973$
   • B$0.098$
   • C$0.126$
   • D$0.0273$

   (c)

   Use the Poisson approximation to find the probability that at most 2 passengers fail to turn up.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A rare genetic condition affects 1 in 500 newborn babies, independently of one another. In one year a region has 1000 births, and $X$ is the number of those babies born with the condition.

   (a)

   State the exact distribution of $X$, name a suitable approximating distribution, and give a reason why the approximation is suitable.

   [3 marks]

   (b)

   (i) Use your approximating distribution to find $P(X\ge4)$.
   (ii) The exact binomial value of $P(X\ge4)$ is $0.1427$ (4 decimal places). Comment on the accuracy of the approximation.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A factory packs screws into boxes of 80. Each screw is independently defective with probability 0.05, and $X$ is the number of defective screws in a box, so $X\sim B(80,0.05)$.

   (a)

   (i) Explain why $X$ can be approximated by a Poisson distribution.
   (ii) Use a Poisson approximation to find the probability that a box contains at least 2 and at most 5 defective screws.

   [6 marks]

   (b)

   A box is rejected if it contains 7 or more defective screws. Use the Poisson approximation to answer (i) and (ii).
   (i) Find the probability that a box is rejected.
   (ii) Find the probability that none of 10 boxes, chosen independently, is rejected.
   (iii) The exact probability that a box is rejected is $0.1053$. State whether the approximation overestimates or underestimates the probability of rejection, and what this means for the factory's planning.

   [6 marks]

   Total for question 4: 12 marks

End of questions