Moderate -0.5 This is a straightforward application of the Poisson approximation to the binomial distribution with clear parameters (n=100000, p=1/25000, so λ=4). The calculation requires finding P(X<2)=P(X=0)+P(X=1) using the standard Poisson formula, which is a routine procedure slightly easier than average due to minimal computational steps and direct application of a standard approximation technique.
1 On average 1 in 25000 people have a rare blood condition. Use a suitable approximating distribution to find the probability that fewer than 2 people in a random sample of 100000 have the condition.
M1 for P(0 or 1) using Poisson, any \(\lambda\); expression of correct form correct \(\lambda\) (allow 1 end error)
\(= 0.0916\) (3 s.f.)
A1 [3]
SR: Use of Bin(100000, 1/25000) scores M1 for P(0,1); allow one end error. A1 0.0916
## Question 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $e^{-4}(1+4)$ | M1, M1 | M1 for P(0 or 1) using Poisson, any $\lambda$; expression of correct form correct $\lambda$ (allow 1 end error) |
| $= 0.0916$ (3 s.f.) | A1 [3] | SR: Use of Bin(100000, 1/25000) scores M1 for P(0,1); allow one end error. A1 0.0916 |
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1 On average 1 in 25000 people have a rare blood condition. Use a suitable approximating distribution to find the probability that fewer than 2 people in a random sample of 100000 have the condition.
\hfill \mbox{\textit{CAIE S2 2014 Q1 [3]}}