| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conditional Probability |
| Type | Independence test requiring preliminary calculations |
| Difficulty | Standard +0.3 This is a standard S1 conditional probability question requiring systematic use of Venn diagram logic and set notation. While it involves multiple steps (finding P(A∩B), then P(A) and P(B), then conditional probability, then independence test), each step follows directly from standard formulas with no conceptual surprises. Slightly above average difficulty due to the preliminary calculations needed before testing independence, but well within typical S1 scope. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| Venn diagram with correct structure; values 0.32, 0.11 and \(A\), \(B\) labelled; 0.22, 0.35 in correct regions | M1, A1, A1 | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(A) = 0.32 + 0.22 = 0.54\); \(P(B) = 0.33\) | M1A1ft; A1ft | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(A \mid B') = \frac{P(A \cap B')}{P(B')} = \frac{32}{67}\) | M1A1 | awrt 0.478 (2) |
| Answer | Marks | Guidance |
|---|---|---|
| For independence \(P(A \cap B) = P(A)P(B)\); for these data \(0.22 \neq 0.54 \times 0.33 = 0.1782\) | M1A1ft | |
| OR \(P(A \mid B') \neq P(A)\) OR \(\frac{2}{3} = P(A \mid B) \neq P(A) = 0.54\) | A1ft | |
| \(\therefore\) NOT independent | (3) |
# Question 6:
## Part (a)
| Venn diagram with correct structure; values 0.32, 0.11 and $A$, $B$ labelled; 0.22, 0.35 in correct regions | M1, A1, A1 | (3) |
## Part (b)
| $P(A) = 0.32 + 0.22 = 0.54$; $P(B) = 0.33$ | M1A1ft; A1ft | (3) |
## Part (c)
| $P(A \mid B') = \frac{P(A \cap B')}{P(B')} = \frac{32}{67}$ | M1A1 | awrt 0.478 (2) |
## Part (d)
| For independence $P(A \cap B) = P(A)P(B)$; for these data $0.22 \neq 0.54 \times 0.33 = 0.1782$ | M1A1ft | |
| OR $P(A \mid B') \neq P(A)$ OR $\frac{2}{3} = P(A \mid B) \neq P(A) = 0.54$ | A1ft | |
| $\therefore$ NOT independent | | (3) |
---6. For the events $A$ and $B$,
$$\mathrm { P } \left( A \cap B ^ { \prime } \right) = 0.32 , \mathrm { P } \left( A ^ { \prime } \cap B \right) = 0.11 \text { and } \mathrm { P } ( A \cup B ) = 0.65$$
\begin{enumerate}[label=(\alph*)]
\item Draw a Venn diagram to illustrate the complete sample space for the events $A$ and $B$.
\item Write down the value of $\mathrm { P } ( A )$ and the value of $\mathrm { P } ( B )$.
\item Find $\mathrm { P } \left( A \mid B ^ { \prime } \right)$.
\item Determine whether or not $A$ and $B$ are independent.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2006 Q6 [11]}}