Standard +0.3 This is a standard conditional probability problem requiring systematic application of the law of total probability and basic probability rules. While it involves multiple conditions and requires careful bookkeeping across several steps, the mathematical techniques are routine for S1 level—setting up a probability tree, using complement rules, and solving a linear equation. The 3-mark allocation confirms it's a straightforward application question rather than requiring novel insight.
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 \(x\). [3]
Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored (isw).
4
(1 – x) × 0.7 × 0.9 = 0.36
M1
B1
(1-x)×0.7×0.9=0.36, (1-x)×0.63=0.36,
0.36
0.63 – 0.63x = 0.36 or 1−x= seen.
0.63
Condone recovery from omission of brackets.
3
x=
Answer
Marks
Guidance
7
A1
Accept 0.428571 to at least 3 sf.
Condone 0.4285 rounding to 0.429 .
3
If M0 awarded, SC B1 for x= or 0.428571 to at least 3 sf.
7
3
Answer
Marks
Guidance
Question
Answer
Marks
Question 4:
4 | Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored (isw).
4 | (1 – x) × 0.7 × 0.9 = 0.36 | M1 | ( 1−x )ab=0.36, a=0.7 or 0.3, b=0.9 or 0.1
B1 | (1-x)×0.7×0.9=0.36, (1-x)×0.63=0.36,
0.36
0.63 – 0.63x = 0.36 or 1−x= seen.
0.63
Condone recovery from omission of brackets.
3
x=
7 | A1 | Accept 0.428571 to at least 3 sf.
Condone 0.4285 rounding to 0.429 .
3
If M0 awarded, SC B1 for x= or 0.428571 to at least 3 sf.
7
3
Question | Answer | Marks | Guidance
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 $x$. [3]
\hfill \mbox{\textit{CAIE S1 2023 Q4 [3]}}