| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conditional Probability |
| Type | Given conditional, find joint or marginal |
| Difficulty | Moderate -0.3 This is a standard S1 conditional probability question testing direct application of formulas: P(A∩B) = P(A|B)×P(B), P(B'|A) = 1 - P(B|A), and P(A'∪B) using complement/De Morgan's laws. Part (d) requires comparing P(A∩B) with P(A)×P(B). All parts are routine bookwork applications with no problem-solving insight needed, making it slightly easier than average, though the multi-part structure and algebraic manipulation keep it close to typical difficulty. |
| Spec | 2.03a Mutually exclusive and independent events2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(B) \times P(A \mid B) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}\) | M1 A1 | |
| \(\frac{P(B' \cap A)}{P(A)} = \frac{\frac{16-1}{...}}{...} = \frac{3}{5}\) | M2 A1 | |
| \((1 - \frac{5}{16}) + \frac{1}{8} = \frac{13}{16}\) | M1 A1 | |
| \(P(A) \times P(B) = \frac{5}{16} \times \frac{1}{2} = \frac{5}{32}\) | M1 | |
| \(\neq P(A \cap B) \therefore\) not independent | M1 A1 | (10) |
| $P(B) \times P(A \mid B) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$ | M1 A1 | |
| $\frac{P(B' \cap A)}{P(A)} = \frac{\frac{16-1}{...}}{...} = \frac{3}{5}$ | M2 A1 | |
| $(1 - \frac{5}{16}) + \frac{1}{8} = \frac{13}{16}$ | M1 A1 | |
| $P(A) \times P(B) = \frac{5}{16} \times \frac{1}{2} = \frac{5}{32}$ | M1 | |
| $\neq P(A \cap B) \therefore$ not independent | M1 A1 | (10) |The events $A$ and $B$ are such that
$$\text{P}(A) = \frac{5}{16}, \text{P}(B) = \frac{1}{2} \text{ and P}(A|B) = \frac{1}{4}$$
Find
\begin{enumerate}[label=(\alph*)]
\item P$(A \cap B)$. [2]
\item P$(B'|A)$. [3]
\item P$(A' \cup B)$. [2]
\item Determine, with a reason, whether or not the events $A$ and $B$ are independent. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q2 [10]}}