| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conditional Probability |
| Type | Bayes with sampling without replacement |
| Difficulty | Standard +0.8 This is a multi-part conditional probability question requiring understanding of probability trees, conditional probability formulas (including Bayes' theorem for part c), and careful tracking of changing sample spaces as cards transfer between packs. While the individual calculations are methodical, the question demands sustained accuracy across multiple parts with 14 total marks, and part (c) requiring reverse conditional probability is non-trivial for S1 level. |
| Spec | 2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(P(A_2) = \frac{7}{30}\) | B1 | |
| (b) \(P(A_1 \cap B_1) = \frac{5}{30} \times \frac{8}{32} = \frac{1}{24}\) | M1 A1 A1 | |
| (c) \(P(B_2) = \left(\frac{3}{30} \times \frac{5}{32}\right) + \left(\frac{27}{30} \times \frac{4}{32}\right) = \frac{41}{320}\), \(P(A_4 | B_2) = \frac{3}{30} \times \frac{5}{32} \times \frac{320}{41} = \frac{4}{41}\) | M1 M1 A1 M1 A1 |
| (d) \(P(A_1 \cup B_2) = \frac{1}{6} \cdot \left(\frac{7}{30} \times \frac{6}{32}\right) \cdot \left(\frac{23}{30} \times \frac{3}{32}\right) = \frac{1}{6} \cdot \frac{7}{160} + \frac{23}{192} = \frac{317}{960}\) or 0.330 | M1 A1 M1 A1 A1 | Total: 14 |
(a) $P(A_2) = \frac{7}{30}$ | B1 |
(b) $P(A_1 \cap B_1) = \frac{5}{30} \times \frac{8}{32} = \frac{1}{24}$ | M1 A1 A1 |
(c) $P(B_2) = \left(\frac{3}{30} \times \frac{5}{32}\right) + \left(\frac{27}{30} \times \frac{4}{32}\right) = \frac{41}{320}$, $P(A_4 | B_2) = \frac{3}{30} \times \frac{5}{32} \times \frac{320}{41} = \frac{4}{41}$ | M1 M1 A1 M1 A1 |
(d) $P(A_1 \cup B_2) = \frac{1}{6} \cdot \left(\frac{7}{30} \times \frac{6}{32}\right) \cdot \left(\frac{23}{30} \times \frac{3}{32}\right) = \frac{1}{6} \cdot \frac{7}{160} + \frac{23}{192} = \frac{317}{960}$ or 0.330 | M1 A1 M1 A1 A1 | **Total: 14**The values of the two variables $A$ and $B$ given in the table in Question 5 are written on cards and placed in two separate packs, which are labelled $A$ and $B$. One card is selected from Pack $A$. Let $A_i$ represent the event that the first digit on this card is $i$.
\begin{enumerate}[label=(\alph*)]
\item Write down the value of P$(A_2)$. [1 mark]
The card taken from Pack $A$ is now transferred into Pack $B$, and one card is picked at random from Pack $B$. Let $B_i$ represent the event that the first digit on this card is $i$.
\item Show that P$(A_1 \cap B_1) = \frac{1}{24}$. [3 marks]
\item Show that P$(A_6 | B_2) = \frac{4}{41}$. [5 marks]
\item Find the value of P$(A_1 \cup B_4)$. [5 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q6 [14]}}