| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2017 |
| Session | January |
| Marks | 9 |
| 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 definitions and calculations. Parts (a)-(d) involve reading values from a given diagram and simple arithmetic. Part (e) tests independence by comparing P(A) with P(A|C), and part (f) requires conditional probability—all standard textbook exercises with no problem-solving insight needed. |
| Spec | 2.03a Mutually exclusive and independent events2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| Part | Answer/Working | Marks |
| (a) | (The event that) the integer selected is prime and ends in a 3 (and is between 1 and 50 inclusive) | B1 |
| (b) | \(\frac{15}{50}\) (or equivalent e.g. 0.30) [condone 30%] | B1 |
| (c) | \(\frac{12}{50}\) (or equivalent e.g. 0.24) [condone 24%] | B1 |
| (d) | \([P(A \mid C) =] \frac{P(A \cap C)}{P(C)} = \frac{\frac{7}{50}}{\frac{30}{50}} = \underline{\frac{7}{30}}\) | M1, A1 |
| (e) | \(\frac{15}{50} \neq \frac{7}{30}\), so not independent. | M1, A1 |
| (f) | \([P(B \mid (A \cap C)) =] \frac{P(B \cap A \cap C)}{P(A \cap C)} = \frac{\frac{2}{50}}{\frac{7}{50}} = \underline{\frac{2}{7}}\) | M1, A1 |
| M1, A1 (2) | ||
| [9 marks] | ||
| SC | For a correct test using correctly labelled \(P(A) = \frac{15}{50}\), \(P(C) = \frac{30}{50}\) and \(P(A \cap C) = \frac{7}{50}\) with all correct probabilities and \(\frac{15}{50} \times \frac{30}{50} = \frac{9}{50} \neq \frac{7}{50}\) (o.e.) seen leading to "not independent" score M0A1 |
| Part | Answer/Working | Marks | Guidance |
|------|---|---|---|
| (a) | (The event that) the integer selected is prime and ends in a 3 (and is between 1 and 50 inclusive) | B1 | (1) |
| (b) | $\frac{15}{50}$ (or equivalent e.g. 0.30) [condone 30%] | B1 | (1) |
| (c) | $\frac{12}{50}$ (or equivalent e.g. 0.24) [condone 24%] | B1 | (1) |
| (d) | $[P(A \mid C) =] \frac{P(A \cap C)}{P(C)} = \frac{\frac{7}{50}}{\frac{30}{50}} = \underline{\frac{7}{30}}$ | M1, A1 | M1 for a correct ratio expression (may be in words) with at least one correct probability substituted or correct ratio expression and $\frac{7}{n}$ or $\frac{m}{30}$ where $7 < n$ or $m < 30$ or fully correct ratio of probabilities. A1 for $\frac{7}{30}$ or any exact equivalent e.g. 0.23 but 0.233 is M1A0 (Correct ans. only = M1A1) |
| (e) | $\frac{15}{50} \neq \frac{7}{30}$, so not independent. | M1, A1 | M1 for correctly comparing 'their (b)' with 'their (d)', can be in words or symbols e.g. $P(A) \neq P(A \mid C)$ in symbols. A1 dependent on a correct (b) and (d) (or awrt 0.233 in (d)) and for concluding not independent. |
| (f) | $[P(B \mid (A \cap C)) =] \frac{P(B \cap A \cap C)}{P(A \cap C)} = \frac{\frac{2}{50}}{\frac{7}{50}} = \underline{\frac{2}{7}}$ | M1, A1 | M1 for a correct ratio expression (may be in words) with at least one correct probability substituted or correct ratio expression and $\frac{r}{7}$ or $\frac{2}{t}$ where $r < 7$ or $2 < t$ or fully correct ratio of probabilities. A1 for $\frac{2}{7}$ or an exact equivalent. Allow awrt 0.286 here as well. (Correct ans. only = M1A1) |
| | | M1, A1 (2) | |
| | | [9 marks] | |
| **SC** | For a correct test using correctly labelled $P(A) = \frac{15}{50}$, $P(C) = \frac{30}{50}$ and $P(A \cap C) = \frac{7}{50}$ with all correct probabilities and $\frac{15}{50} \times \frac{30}{50} = \frac{9}{50} \neq \frac{7}{50}$ (o.e.) seen leading to "not independent" score M0A1 | | |\begin{enumerate}
\item An integer is selected at random from the integers 1 to 50 inclusive.\\
$A$ is the event that the integer selected is prime.\\
$B$ is the event that the integer selected ends in a 3\\
$C$ is the event that the integer selected is greater than 20\\
The Venn diagram shows the number of integers in each region for the events $A , B$ and $C$\\
\includegraphics[max width=\textwidth, alt={}, center]{1130517e-33d0-41b1-9303-2d981379954d-04_607_1125_593_413}\\
(a) Describe in words the event $( A \cap B )$\\
(b) Write down the probability that the integer selected is prime.\\
(c) Find $\mathrm { P } \left( [ A \cup B \cup C ] ^ { \prime } \right)$
\end{enumerate}
Given that the integer selected is greater than 20\\
(d) find the probability that it is prime.
Using your answers to (b) and (d),\\
(e) state, with a reason, whether or not the events $A$ and $C$ are statistically independent.
Given that the integer selected is greater than 20 and prime,\\
(f) find the probability that it ends in a 3
\hfill \mbox{\textit{Edexcel S1 2017 Q2 [9]}}