Standard +0.3 This is a straightforward application of normal approximation to Poisson (λ=4.5), requiring only identification of the appropriate approximation, continuity correction, and standard normal table lookup. The calculation is routine with no conceptual challenges beyond recognizing when to use the approximation.
2 A certain machine makes matches. One match in 10000 on average is defective. Using a suitable approximation, calculate the probability that a random sample of 45000 matches will include 2,3 or 4 defective matches.
For using Poisson approximation any mean; For correct mean used; For calculating P(2, 3, 4) their mean; For correct numerical expression; For correct answer
NB: Use of Normal can score B1 M1; SR: Correct Bin scores M1 A1 A1 only
[5]
$\lambda = 4.5$
$P(X = 2, 3, 4) = e^{-4.5}\left(\frac{4.5^2}{2!} + \frac{4.5^3}{3!} + \frac{4.5^4}{4!}\right) = 0.471$ | M1, B1, M1, A1, A1 | For using Poisson approximation any mean; For correct mean used; For calculating P(2, 3, 4) their mean; For correct numerical expression; For correct answer
NB: Use of Normal can score B1 M1; SR: Correct Bin scores M1 A1 A1 only | [5]
2 A certain machine makes matches. One match in 10000 on average is defective. Using a suitable approximation, calculate the probability that a random sample of 45000 matches will include 2,3 or 4 defective matches.
\hfill \mbox{\textit{CAIE S2 2003 Q2 [5]}}