A-LevelMathematicsProbability and statistics9709/62 · Oct/Nov 2023 · Q79 Marks

A random variable X has the distribution Po(2.4).

(a)Find P(2 ≤ X < 4).[2]
(b)Two independent values of X are chosen. Find the probability that both of these values are greater than 1.[3]
(c)(i)It is given that P(X = r) < P(X = r + 1). Find the set of possible values of r.[3]
(c)(ii)Hence find the value of r for which P(X = r) is greatest.[1]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

[M1] Allow M1 for e-2.4 (2.4²/2! + 2.4³/3! + 2.4⁴/4!). Expression must be seen. e-2.4 (2.4²/2! + 2.4³/3!) = e-2.4 (2.88 + 2.304) = 0.2613+0.2090
[A1] SC B1 0.47(0) with no working. = 0.47(0)
[M1] Allow one end error. Allow any λ. 1-e-2.4(1 + 2.4) [= 0.691558]
[M1] Squaring their probability (λ ≠ 4.8). (1-e-2.4(1 + 2.4))²
[A1] SC B2 0.478 with no working. = 0.478 (3 sf)
[M1] For both expressions seen. e-2.4 x 2.4^r / r! < e-2.4 x 2.4^(r+1) / (r+1)!
[A1*] Or r < 1.4 (must have correct inequality). r + 1 < 2.4
[DA1] Set is r = 0, 1
[B1] r = 2

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Topic: Probability and statistics, A-Level Mathematics (syllabus 9709). Structured question, 9 marks.
Source: A-Level Mathematics, Oct/Nov 2023 Paper 6 Variant 2, question 7.

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