A-LevelMathematicsProbability and statistics9709/61 · May/June 2021 · Q57 Marks

On average, 1 in 75000 adults has a certain genetic disorder.

(a)Use a suitable approximating distribution to find the probability that, in a random sample of 10000 people, at least 1 has the genetic disorder.[3]
(b)In a random sample of n people, where n is large, the probability that no-one has the genetic disorder is more than 0.9. Find the largest possible value of n.[4]

Mark scheme

Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.

[M1] For Poisson approximation parameter `λ = (1/75000) * 10000 = 10000/75000 = 10/75 = 2/15`. SOI. Allow `Po(0.133)`.
[M1] For calculating `P(X >= 1) = 1 - P(X=0) = 1 - e^(-λ)`. Allow incorrect λ, allow one end error.
[A1] Final probability `= 0.125 (3 sf)`. SC Partially unsupported final answer: `Po(2/15)` stated B1 then unsupported `0.125` B1. SC Use of Binomial (`0.1248`) B1 only. Use of Normal scores M0.
[B1] For Poisson parameter `λ = n/75000`.
[M1] For inequality `e^(-n/75000) > 0.9`. Allow '='.
[M1] For taking natural logarithm on both sides: `-n/75000 > ln 0.9` [leading to `n < 7902.04`].
[A1] Largest possible value of n is `7902`. CWO. Must be an integer.

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