| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2017 |
| Session | October |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Definitions |
| Type | Venn diagram completion |
| Difficulty | Moderate -0.8 This is a routine S1 Venn diagram question requiring systematic application of basic probability rules (mutually exclusive events, independence, addition rule, conditional probability). While multi-part with 6 marks worth of work, each step follows standard procedures with no novel insight required—identifying mutually exclusive events from a diagram, using P(A∩C)=P(A)×P(C) for independence, then algebraic manipulation to find remaining probabilities. Easier than average A-level due to being purely procedural recall and calculation. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | B and C (only) | B1 |
| (b) | \(P(A \cap C) = 0.6 \times 0.35\) so \([w =] \mathbf{0.21}\) | B1cso |
| (c) | \(x = P(C) - w = \mathbf{0.14}\) | B1 |
| \(y = P(A) - w - P(B) y = \mathbf{0.24}\) | M1, A1 | M1 for a correct expression for \(y\) |
| \(z = 1 - P(A \cup C) = \mathbf{0.26}\) | B1ft | A1 for \(y = 0.24\) 2nd B1ft for \(z = 0.26\) or correct ft of their values to make sum = 1 (provided all probs) These values may be seen in incorrect regions in the Venn diagram |
| (d) | \([x + y =] \mathbf{0.38}\) | B1ft |
| (e) | \([P(B \cup C) = 0.15 + 0.35] = \mathbf{0.5}\) | B1cao |
| (f) | \(\left[P\left(A\mid[B \cup C]\right)\right] = \frac{P(A \cap [B \cup C])}{P(B \cup C)} = \frac{0.15 + 0.21}{"0.5"}\) | M1A1ft |
| \(= \mathbf{0.72}\) | A1 | 2nd A1 for 0.72 or exact equivalent e.g. \(\frac{18}{25}\) |
| **(a)** | B and C (only) | B1 | |
|---|---|---|---|
| **(b)** | $P(A \cap C) = 0.6 \times 0.35$ so $[w =] \mathbf{0.21}$ | B1cso | B1cso for 0.21 clearly from $P(A) \times P(C)$ or $0.6 \times 0.35$ and no incorrect statements seen |
| **(c)** | $x = P(C) - w = \mathbf{0.14}$ | B1 | 1st B1 for $x = 0.14$ |
| | $y = P(A) - w - P(B) y = \mathbf{0.24}$ | M1, A1 | M1 for a correct expression for $y$ |
| | $z = 1 - P(A \cup C) = \mathbf{0.26}$ | B1ft | A1 for $y = 0.24$ 2nd B1ft for $z = 0.26$ or correct ft of their values to make sum = 1 (provided all probs) These values may be seen in incorrect regions in the Venn diagram |
| **(d)** | $[x + y =] \mathbf{0.38}$ | B1ft | |
| **(e)** | $[P(B \cup C) = 0.15 + 0.35] = \mathbf{0.5}$ | B1cao | |
| **(f)** | $\left[P\left(A\mid[B \cup C]\right)\right] = \frac{P(A \cap [B \cup C])}{P(B \cup C)} = \frac{0.15 + 0.21}{"0.5"}$ | M1A1ft | M1 for a correct ratio of probabilities formula num of: $P(B \cup C \cap A)$ or $P(A \cap [B \cup C])$ with brackets and some correct probability, ft their (e) May be implied by correct ratio. It their formula is incorrect but numerator is 0.15 + 0.21 and denominator is their (e) Can award M1A1ft for $\frac{0.15 + 0.21}{\text{"their 0.5"}}$ even if their formula is incorrect |
| | $= \mathbf{0.72}$ | A1 | 2nd A1 for 0.72 or exact equivalent e.g. $\frac{18}{25}$ |
**[Total 11]**
---\begin{enumerate}
\item The Venn diagram, where $w , x , y$ and $z$ are probabilities, shows the probabilities of a group of students buying each of 3 magazines.
\end{enumerate}
A represents the event that a student buys magazine $A$ and $\mathrm { P } ( A ) = 0.60$\\
$B$ represents the event that a student buys magazine $B$ and $\mathrm { P } ( B ) = 0.15$\\
$C$ represents the event that a student buys magazine $C$ and $\mathrm { P } ( C ) = 0.35$\\
\includegraphics[max width=\textwidth, alt={}, center]{77ae01cd-2b58-48ab-889f-272e27ecf99d-06_504_755_641_596}\\
(a) State which two of the three events $A$, $B$ and $C$ are mutually exclusive.
The events $A$ and $C$ are independent.\\
(b) Show that $w = 0.21$\\
(c) Find the value of $x$, the value of $y$ and the value of $z$.\\
(d) Find the probability that a student selected at random buys only one of these magazines.\\
(e) Find the probability that a student selected at random buys magazine $B$ or magazine $C$.\\
(f) Find $\mathrm { P } ( A \mid [ B \cup C ] )$
\begin{center}
\end{center}
\hfill \mbox{\textit{Edexcel S1 2017 Q2 [11]}}