| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Definitions |
| Type | Combined event algebra |
| Difficulty | Moderate -0.3 This is a standard S1 probability question testing basic definitions (mutually exclusive, independence, conditional probability) and Venn diagrams. Part (a) uses the addition rule for mutually exclusive events, (b) applies the independence formula P(A∪B) = P(A) + P(B) - P(A)P(B), (c) uses the definition of conditional probability with independence, and (d) requires constructing a Venn diagram from given information. All parts are routine applications of formulas with straightforward algebra, making it slightly easier than average for A-level. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables |
| Answer | Marks | Guidance |
|---|---|---|
| Part | Answer/Working | Marks |
| (a) | \([P(A \cup C)] = \frac{9}{10}\) oe | B1 |
| (b) | \(P\left(A \cup B\right) = P(A) + P(B) - P(A) \times P(B)\) | M1 |
| \(\frac{5}{8} = \frac{2}{5} + P(B) - \frac{2}{5}P(B)\) | M1 A1 | 2nd M1 for use of \(P(A \cap B) = P(A) \times P(B)\) (But just seeing \(\frac{2}{8} \times \frac{3}{5} = \frac{3}{20}\) on its own is M0M0). 1st A1 correct equation |
| \(P(B) = \frac{3}{8}\) * | A1cso | 2nd A1 cso (No wrong working seen dependent on all previous marks) (allow full verification method, however, substitution of P(B)=3/8 into only one P(B) to find the other P(B) (e.g. using 3/20 to find 3/8) can score M1M0A0A0) |
| (c) | \([P(A \mid B)] = P(A) = \frac{2}{5}\) oe | B1 |
| (d) | Diagram | B1 |
| 0.15 and 0.25 | M1 | 1st M1: 0.15 placed in \((A \cap B \cap C')\) and 0.25 placed in \((A \cap B' \cap C')\) |
| 0.05 and 0.05 | M1 | 2nd M1: 0.3 − 'their 0.25' and 1 − ('their 0.15' + 'their 0.25' + 'their 0.05' + \(\frac{1}{4}\)) |
| 0.175 and 0.325 | M1 A1 | 3rd M1: \(\frac{3}{8}\) − ('their 0.15' + 'their 0.05'), i.e. \(P(\overline{B}) = \frac{3}{8}\) and \(\frac{1}{2}\) − 'their 0.175', i.e. \(P(C) = \frac{1}{2}\) |
| For 3rd M1 mark, blank regions inside P(B) and P(C) are not treated as 0s and score M0. A1 fully correct with box |
| Part | Answer/Working | Marks | Guidance |
|------|---|---|---|
| (a) | $[P(A \cup C)] = \frac{9}{10}$ oe | B1 | |
| (b) | $P\left(A \cup B\right) = P(A) + P(B) - P(A) \times P(B)$ | M1 | 1st M1 for use of $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ |
| | $\frac{5}{8} = \frac{2}{5} + P(B) - \frac{2}{5}P(B)$ | M1 A1 | 2nd M1 for use of $P(A \cap B) = P(A) \times P(B)$ (But just seeing $\frac{2}{8} \times \frac{3}{5} = \frac{3}{20}$ on its own is M0M0). 1st A1 correct equation |
| | $P(B) = \frac{3}{8}$ * | A1cso | 2nd A1 cso (No wrong working seen dependent on all previous marks) (allow full verification method, however, substitution of P(B)=3/8 into only one P(B) to find the other P(B) (e.g. using 3/20 to find 3/8) can score M1M0A0A0) |
| (c) | $[P(A \mid B)] = P(A) = \frac{2}{5}$ oe | B1 | |
| (d) | Diagram | B1 | B1: 3 circles intersecting, see diagram below, (at least 2 labelled) with the two zeros showing A does not intersect C (Do not allow blank spaces for the two zeros) |
| | 0.15 and 0.25 | M1 | 1st M1: 0.15 placed in $(A \cap B \cap C')$ and 0.25 placed in $(A \cap B' \cap C')$ |
| | 0.05 and 0.05 | M1 | 2nd M1: 0.3 − 'their 0.25' and 1 − ('their 0.15' + 'their 0.25' + 'their 0.05' + $\frac{1}{4}$) |
| | 0.175 and 0.325 | M1 A1 | 3rd M1: $\frac{3}{8}$ − ('their 0.15' + 'their 0.05'), i.e. $P(\overline{B}) = \frac{3}{8}$ and $\frac{1}{2}$ − 'their 0.175', i.e. $P(C) = \frac{1}{2}$ |
| | | | For 3rd M1 mark, blank regions inside P(B) and P(C) are not treated as 0s and score M0. A1 fully correct with box |
**Total: 11 marks**
---6. Three events $A , B$ and $C$ are such that
$$\mathrm { P } ( A ) = \frac { 2 } { 5 } \quad \mathrm { P } ( C ) = \frac { 1 } { 2 } \quad \mathrm { P } ( A \cup B ) = \frac { 5 } { 8 }$$
Given that $A$ and $C$ are mutually exclusive find
\begin{enumerate}[label=(\alph*)]
\item $\mathrm { P } ( A \cup C )$
Given that $A$ and $B$ are independent
\item show that $\mathrm { P } ( B ) = \frac { 3 } { 8 }$
\item Find $\mathrm { P } ( A \mid B )$
Given that $\mathrm { P } \left( C ^ { \prime } \cap B ^ { \prime } \right) = 0.3$
\item draw a Venn diagram to represent the events $A , B$ and $C$
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2016 Q6 [11]}}