Standard +0.3 This is a straightforward application of the independence condition P(A∩B) = P(A)×P(B) using given percentages. Students must recognize that P(M∩E) = 1 - 0.14 - P(M only) - P(E only), which requires basic set theory and Venn diagram reasoning, but the calculations are routine once the setup is identified. Slightly easier than average due to clear structure and standard technique.
12 You must show detailed reasoning in this question.
In the summer of 2017 in England a large number of candidates sat GCSE examinations in both mathematics and English. 56\% of these candidates achieved at least level 4 in mathematics and \(80 \%\) of these candidates achieved at least level 4 in English. 14\% of these candidates did not achieve at least level 4 in either mathematics or English.
Determine whether achieving level 4 or above in English and achieving level 4 or above in mathematics were independent events.
Use of contingency table or Venn diagram or \(P(A \text{ or } B)=P(A)+P(B)-P(A \text{ and } B)\)
M1 (AO3.1b)
\(0.56\), \(0.8\) and \(0.14\) must be correctly placed; e.g. \(1-0.14=0.56+0.8-P(A \text{ and } B)\); where \(A\) denotes "passing" maths and \(B\) denotes "passing" English; the first M1A1 may be awarded for working with percentages
\(P(A \text{ and } B)=0.5\)
A1 (AO2.1)
Or \(P(A/B)=\frac{0.5}{0.80}\)
\(P(A)\times P(B)=0.56\times0.80\)
M1 (AO1.1)
\(=0.625\) or \(\frac{5}{8}\) seen; allow equivalent argument based on showing \(A'\) and \(B'\) not independent
\(=0.448\) seen
A1 (AO1.1)
\(=0.625\) or \(\frac{5}{8}\neq0.56\)
\(0.448\neq0.5\) or \(0.56\times0.80\neq0.5\) so not independent
A1 (AO3.2a)
[5]
## Question 12:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use of contingency table or Venn diagram or $P(A \text{ or } B)=P(A)+P(B)-P(A \text{ and } B)$ | M1 (AO3.1b) | $0.56$, $0.8$ and $0.14$ must be correctly placed; e.g. $1-0.14=0.56+0.8-P(A \text{ and } B)$; where $A$ denotes "passing" maths and $B$ denotes "passing" English; the first M1A1 may be awarded for working with percentages |
| $P(A \text{ and } B)=0.5$ | A1 (AO2.1) | Or $P(A/B)=\frac{0.5}{0.80}$ |
| $P(A)\times P(B)=0.56\times0.80$ | M1 (AO1.1) | $=0.625$ or $\frac{5}{8}$ seen; allow equivalent argument based on showing $A'$ and $B'$ not independent |
| $=0.448$ seen | A1 (AO1.1) | $=0.625$ or $\frac{5}{8}\neq0.56$ |
| $0.448\neq0.5$ or $0.56\times0.80\neq0.5$ so not independent | A1 (AO3.2a) | |
| **[5]** | | |
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12 You must show detailed reasoning in this question.
In the summer of 2017 in England a large number of candidates sat GCSE examinations in both mathematics and English. 56\% of these candidates achieved at least level 4 in mathematics and $80 \%$ of these candidates achieved at least level 4 in English. 14\% of these candidates did not achieve at least level 4 in either mathematics or English.
Determine whether achieving level 4 or above in English and achieving level 4 or above in mathematics were independent events.
\hfill \mbox{\textit{OCR MEI Paper 2 2018 Q12 [5]}}