CAIE A-Level Mathematics 6.1.4 Poisson Approximation to the Binomial

CAIE A-Level Mathematics 6.1.4 Poisson Approximation to the Binomial
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise replacing a binomial model by Po(np) when trials are numerous and success is rare, justifying the choice and evaluating exact or cumulative probabilities.

How this is tested

  • check that n is large and p is small, using the stated guidance n > 50 and np < 5
  • set the approximating Poisson mean to λ = np and state the complete Po(λ) model
  • translate the binomial event without continuity correction and evaluate its Poisson sum or tail

Question 3

[Maximum number: 6]

The data produced by a certain data entry firm always include a small number of incorrect characters that occur at random. The proportion of incorrect characters is denoted by p, and experience has shown that p=0.0001. A particular data set from the firm contains 14500 characters, of which X characters are incorrect.

Question 3(a)

(a)

Use a suitable approximating distribution to find P(X<4).

The firm's management wishes to decrease the value of p by giving their employees some training. Their aim is that, for a data set containing 14500 characters, the value of P(X=0) for the new value of p should be double the value of P(X=0) when p=0.0001.

[ 3 ]

Question 3(b)

(b)

Use a suitable approximating distribution to find the new value of p.

[ 3 ]