| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2004 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Independent Events |
| Type | Independence with three or more events |
| Difficulty | Standard +0.3 This is a straightforward application of probability rules for mutually exclusive and independent events. Part (a) tests understanding of Venn diagrams, while parts (b)-(d) require standard formulas: P(AC) uses independence, P(A∪B) uses mutual exclusivity, and P(C) follows from the addition rule. All steps are routine with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Venn diagram with \(A\), \(B\), \(C\) inside \(S\) | B1 | \(A\), \(B\), \(C\) inside \(S\) |
| \(A\), \(B\) no overlap shown | B1 | \(A\) and \(B\) must be mutually exclusive (no overlap) |
| \(A\), \(C\) overlap shown | B1 | \(A\) and \(C\) must intersect |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(A\ | C) = \dfrac{P(A \cap C)}{P(C)} = \dfrac{P(A)P(C)}{P(C)} = P(A)\) | M1 |
| \(= 0.2\) | A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\) | M1 | Use of \(P(A \cup B)\); \(P(A \cap B) = 0\) can be implied |
| \(= 0.2 + 0.4 - 0\) | ||
| \(= 0.6\) | A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(A \cup C) = P(A) + P(C) - P(A \cap C)\) | M1 | Use of \(P(A \cup C)\) and independence |
| \(\therefore 0.7 = 0.2 + P(C) - 0.2\,P(C)\) | A1 | |
| \(\therefore 0.5 = P(C)\{1 - 0.2\}\) | M | Solving for \(P(C)\) from an equation with \(2P(C)\) terms |
| \(\therefore P(C) = \dfrac{5}{8}\) | A1 | (4 marks) |
# Question 6:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| Venn diagram with $A$, $B$, $C$ inside $S$ | B1 | $A$, $B$, $C$ inside $S$ |
| $A$, $B$ no overlap shown | B1 | $A$ and $B$ must be mutually exclusive (no overlap) |
| $A$, $C$ overlap shown | B1 | $A$ and $C$ must intersect | **(3 marks)** |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(A\|C) = \dfrac{P(A \cap C)}{P(C)} = \dfrac{P(A)P(C)}{P(C)} = P(A)$ | M1 | Use of independence |
| $= 0.2$ | A1 | **(2 marks)** |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ | M1 | Use of $P(A \cup B)$; $P(A \cap B) = 0$ can be implied |
| $= 0.2 + 0.4 - 0$ | | |
| $= 0.6$ | A1 | **(2 marks)** |
## Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(A \cup C) = P(A) + P(C) - P(A \cap C)$ | M1 | Use of $P(A \cup C)$ and independence |
| $\therefore 0.7 = 0.2 + P(C) - 0.2\,P(C)$ | A1 | |
| $\therefore 0.5 = P(C)\{1 - 0.2\}$ | M | Solving for $P(C)$ from an equation with $2P(C)$ terms |
| $\therefore P(C) = \dfrac{5}{8}$ | A1 | **(4 marks)** |
**NB** $P(B \cup C) = P(B) + P(C) - P(B \cap C) = 0.4 + 0.625 - P(B \cap C) \Rightarrow P(B \cap C) > 0$
**(11 marks total)**6. Three events $A , B$ and $C$ are defined in the sample space $S$. The events $A$ and $B$ are mutually exclusive and $A$ and $C$ are independent.
\begin{enumerate}[label=(\alph*)]
\item Draw a Venn diagram to illustrate the relationships between the 3 events and the sample space.
Given that $\mathrm { P } ( A ) = 0.2 , \mathrm { P } ( B ) = 0.4$ and $\mathrm { P } ( A \cup C ) = 0.7$, find
\item $\mathrm { P } ( A C )$,
\item $\mathrm { P } ( A \cup B )$,
\item $\mathrm { P } ( C )$.
END
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2004 Q6 [11]}}