A-LevelMathematicsProbability and statistics9709/62 · Oct/Nov 2020 · Q710 Marks

Before a certain type of book is published it is checked for errors, which are then corrected. For costing purposes each error is classified as either minor or major. The numbers of minor and major errors in a book are modelled by the independent distributions N(380, 140) and N(210, 80) respectively. You should assume that no continuity corrections are needed when using these models. A book of this type is chosen at random.

(a)Find the probability that the number of minor errors is at least 200 more than the number of major errors.[5]
(b)The costs of correcting a minor error and a major error are 20 cents and 50 cents respectively. Find the probability that the total cost of correcting the errors in the book is less than $190.[5]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.

[B1] E(M-J) = 380 - 210 = 170 (seen or implied)
[B1] Var(M-J) = 140 + 80 = 220 (seen or implied)
[M1] Standardise: Z = (200 - 170) / sqrt(220)
[A1] Z = 2.023
[M1] Calculate probability P(Z > 2.023) = 1 - Φ(2.023)
[A1] = 0.0216 (3sf)
[B1] E(Total Cost) = 380 × 20 + 210 × 50 = 18100 cents or $181 (seen or implied)
[B1] Var(Total Cost) = 140 × 20² + 80 × 50² = 256000 cents² or 25.6 dollar² (seen or implied)
[M1] Standardise: Z = (19000 - 18100) / sqrt(256000) or ($190 - $181) / sqrt(25.6)
[A1] Z = 1.778
[M1] Calculate probability P(Z < 1.778) = Φ(1.778)
[A1] = 0.962 or 0.963 (3sf)

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