A-LevelMathematicsProbability and statistics9709/52 · Feb/Mar 2023 · Q43 Marks

The probability that it will rain on any given day is x. If it is raining, the probability that Aran wears a hat is 0.8 and if it is not raining, the probability that he wears a hat is 0.3. Whether it is raining or not, if Aran wears a hat, the probability that he wears a scarf is 0.4. If he does not wear a hat, the probability that he wears a scarf is 0.1. The probability that on a randomly chosen day it is not raining and Aran is not wearing a hat or a scarf is 0.36. Find the value of x.

Mark scheme

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

[M1] (1-x)×a×b=0.36, a=0.7 or 0.3, b=0.9or0.1
[B1] (1-x)×0.7×0.9=0.36, (1-x)×0.63=0.36, 0.63 - 0.63x = 0.36 or 1-x=0.36/0.63 seen. Condone recovery from omission of brackets.
[A1] x=3/7. Accept 0.428571 to at least 3 sf. Condone 0.4285 rounding to 0.429. If M0 awarded, SC B1 for x = 3/7 or 0.428571 to at least 3 sf.

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Topic: Probability and statistics, A-Level Mathematics (syllabus 9709). Structured question, 3 marks.
Source: A-Level Mathematics, Feb/Mar 2023 Paper 5 Variant 2, question 4.

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