| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2022 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Definitions |
| Type | Venn diagram completion |
| Difficulty | Easy -1.2 This is a straightforward S1 Venn diagram question requiring only basic probability calculations (reading values from diagram, applying complement/intersection/union rules, and computing an expectation). All parts follow standard textbook patterns with no problem-solving insight needed—purely mechanical application of definitions once the diagram is interpreted. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space |
| Answer | Marks | Guidance |
|---|---|---|
| Part | Scheme | Marks |
| (a) | \(P(C') = \frac{103}{120}\) or equivalent | B1 (1) |
| (b) | \(P(A \cap B \cap C') = 0\) | B1 (1) |
| (c) | \(P(A \cup B \cup C') = \frac{9+3+2+5+1+93}{120}\) or \(P(A \cup B \cup C') = 1 - \frac{7}{120} = \frac{113}{120}\) or equivalent | M1, A1 (2) |
| (d) | \(P(\text{At most 1}) = P(0 \text{ or } 1) = \frac{93+9+7+1}{120}\) or \(\frac{120-2-5-3}{120} = \frac{110}{120}\) or equivalent | M1, A1 (2) |
| (e) | \(P(A \mid \text{At most 1}) = \frac{9/120}{110/120} = \frac{9}{110}\) or equivalent | M1, A1 (2) |
| (f) | \(P(X=0) = \frac{93}{120}\); \(P(X=1) = \frac{17}{120}\); \(P(X=2) = \frac{8}{120}\); \(P(X=3) = \frac{2}{120}\) | M1 |
| \(E(X) = \left[\frac{93}{120} \times 0\right] + \frac{17}{120} \times 1 + \frac{8}{120} \times 2 + \frac{2}{120} \times 3 = \frac{13}{40}\) or 0.325 or equivalent | M1, A1 (3) | 1st M1 for probability distribution (condone missing \(P(X=0)\)) awrt 0.14, awrt 0.067 and awrt 0.017. May be implied by correct expression for \(E(X)\). At least 2 correct must be associated with correct \(x\) value. 2nd M1 correct follow through expression for \(E(X)\) if their probabilities and \(X\) values. A1 Dep on both previous method marks being awarded. Working must be checked. A correct answer with no working scores 3/3. Special cases: \(P(X=17) = 17/120\) (awrt 0.14); \(P(X=8) = 8/120\) (awrt 0.067); \(P(X=14) = 14/120\) (awrt 0.12) leading to awrt 4.58 or \(183/40\) gains M0M1A0 |
| Part | Scheme | Marks | Guidance |
|------|--------|-------|----------|
| (a) | $P(C') = \frac{103}{120}$ or equivalent | B1 (1) | Allow awrt 0.858 |
| (b) | $P(A \cap B \cap C') = 0$ | B1 (1) | |
| (c) | $P(A \cup B \cup C') = \frac{9+3+2+5+1+93}{120}$ or $P(A \cup B \cup C') = 1 - \frac{7}{120} = \frac{113}{120}$ or equivalent | M1, A1 (2) | Allow awrt 0.942 |
| (d) | $P(\text{At most 1}) = P(0 \text{ or } 1) = \frac{93+9+7+1}{120}$ or $\frac{120-2-5-3}{120} = \frac{110}{120}$ or equivalent | M1, A1 (2) | Allow awrt 0.917 |
| (e) | $P(A \mid \text{At most 1}) = \frac{9/120}{110/120} = \frac{9}{110}$ or equivalent | M1, A1 (2) | Allow awrt 0.0818 |
| (f) | $P(X=0) = \frac{93}{120}$; $P(X=1) = \frac{17}{120}$; $P(X=2) = \frac{8}{120}$; $P(X=3) = \frac{2}{120}$ | M1 | |
| | $E(X) = \left[\frac{93}{120} \times 0\right] + \frac{17}{120} \times 1 + \frac{8}{120} \times 2 + \frac{2}{120} \times 3 = \frac{13}{40}$ or 0.325 or equivalent | M1, A1 (3) | 1st M1 for probability distribution (condone missing $P(X=0)$) awrt 0.14, awrt 0.067 and awrt 0.017. May be implied by correct expression for $E(X)$. At least 2 correct must be associated with correct $x$ value. 2nd M1 correct follow through expression for $E(X)$ if their probabilities and $X$ values. A1 Dep on both previous method marks being awarded. Working must be checked. A correct answer with no working scores 3/3. Special cases: $P(X=17) = 17/120$ (awrt 0.14); $P(X=8) = 8/120$ (awrt 0.067); $P(X=14) = 14/120$ (awrt 0.12) leading to awrt 4.58 or $183/40$ gains M0M1A0 |
---\begin{enumerate}
\item A factory produces shoes.
\end{enumerate}
A quality control inspector at the factory checks a sample of 120 shoes for each of three types of defect. The Venn diagram represents the inspector's results.
A represents the event that a shoe has defective stitching $B$ represents the event that a shoe has defective colouring $C$ represents the event that a shoe has defective soles\\
\includegraphics[max width=\textwidth, alt={}, center]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-02_684_935_607_566}
One of the shoes in the sample is selected at random.\\
(a) Find the probability that it does not have defective soles.\\
(b) Find $\mathrm { P } \left( A \cap B \cap C ^ { \prime } \right)$\\
(c) Find $\mathrm { P } \left( A \cup B \cup C ^ { \prime } \right)$\\
(d) Find the probability that the shoe has at most one type of defect.\\
(e) Given the selected shoe has at most one type of defect, find the probability it has defective stitching.
The random variable $X$ is the number of the events $A , B , C$ that occur for a randomly selected shoe.\\
(f) Find $\mathrm { E } ( X )$
\section*{This is a copy of the Venn diagram for this question.}
\includegraphics[max width=\textwidth, alt={}, center]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-05_684_940_388_566}\\
\hfill \mbox{\textit{Edexcel S1 2022 Q1 [11]}}