| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2022 |
| Session | October |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conditional Probability |
| Type | Conditional probability with algebraic expressions |
| Difficulty | Standard +0.3 This is a standard S1 conditional probability question using a Venn diagram with algebraic expressions. Part (a) requires reading probabilities directly from the diagram. Parts (b) and (c) involve setting up and solving simple linear equations using conditional probability formulas P(A|B) = P(A∩B)/P(B) and set operations. While it requires careful bookkeeping across multiple parts, the techniques are routine for S1 and involve no novel problem-solving—just systematic application of definitions. Slightly easier than average due to the structured, step-by-step nature. |
| 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 |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \([P(A) =]\ \mathbf{0.25}\) | B1 | 0.25 oe |
| \([P(A\ | B) =]\ \mathbf{1}\) | B1 |
| \([P(A\ | C) =]\ \mathbf{0}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\frac{q}{q+r} = \frac{3}{5}\) | M1 | Correct expression for \(P(C\ |
| \(0.13 + p + s = \frac{7}{10}\) | M1 | Correct expression for \(P(B' \cap D') = \frac{7}{10}\) |
| \(p + q + r + s + 0.12 + 0.13 = 1\) | M1 | Correct equation using sum of probabilities \(= 1\) |
| \(\frac{q}{0.3-0.12} = \frac{3}{5}\) or \(0.3 = 0.12 + 1.5r + r\) or \(0.3 = 0.12 + q + \frac{2}{3}q\) | dM1 | Dep on all 3 previous M1; solving to obtain correct equation in single variable |
| \(q = \mathbf{0.108}\) | A1 | \(q = 0.108\) or \(\frac{27}{250}\) oe |
| \(r = \mathbf{0.072}\) | A1 | \(r = 0.072\) or \(\frac{9}{125}\) oe |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\frac{5}{8} = 0.13 + 0.12 + \text{'0.072'} + s\) oe | M1 | Correct expression for \(P(B \cup C') = \frac{5}{8}\); ft their value for \(r\); allow use of letter \(r\) |
| \(s = \mathbf{0.303}\) | A1 | \(s = 0.303\) oe |
# Question 6:
## Part (a)(i)–(iii):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $[P(A) =]\ \mathbf{0.25}$ | B1 | 0.25 oe |
| $[P(A\|B) =]\ \mathbf{1}$ | B1 | 1 cao |
| $[P(A\|C) =]\ \mathbf{0}$ | B1 | 0 cao |
## Part (b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\frac{q}{q+r} = \frac{3}{5}$ | M1 | Correct expression for $P(C\|D) = \frac{3}{5}$; allow $P(D)$ for $q+r$ |
| $0.13 + p + s = \frac{7}{10}$ | M1 | Correct expression for $P(B' \cap D') = \frac{7}{10}$ |
| $p + q + r + s + 0.12 + 0.13 = 1$ | M1 | Correct equation using sum of probabilities $= 1$ |
| $\frac{q}{0.3-0.12} = \frac{3}{5}$ or $0.3 = 0.12 + 1.5r + r$ or $0.3 = 0.12 + q + \frac{2}{3}q$ | dM1 | Dep on all 3 previous M1; solving to obtain correct equation in single variable |
| $q = \mathbf{0.108}$ | A1 | $q = 0.108$ or $\frac{27}{250}$ oe |
| $r = \mathbf{0.072}$ | A1 | $r = 0.072$ or $\frac{9}{125}$ oe |
## Part (c):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\frac{5}{8} = 0.13 + 0.12 + \text{'0.072'} + s$ oe | M1 | Correct expression for $P(B \cup C') = \frac{5}{8}$; ft their value for $r$; allow use of letter $r$ |
| $s = \mathbf{0.303}$ | A1 | $s = 0.303$ oe |
---\begin{enumerate}
\item The Venn diagram shows the events $A , B , C$ and $D$, where $p , q , r$ and $s$ are probabilities.\\
\includegraphics[max width=\textwidth, alt={}, center]{1fda59cb-059e-4850-810f-cc3e69bc058e-20_504_826_296_621}\\
(a) Write down the value of\\
(i) $\mathrm { P } ( A )$\\
(ii) $\mathrm { P } ( A \mid B )$\\
(iii) $\mathrm { P } ( A \mid C )$
\end{enumerate}
Given that $\mathrm { P } \left( B ^ { \prime } \cap D ^ { \prime } \right) = \frac { 7 } { 10 }$ and $\mathrm { P } ( C \mid D ) = \frac { 3 } { 5 }$\\
(b) find the exact value of $q$ and the exact value of $r$
Given also that $\mathrm { P } \left( B \cup C ^ { \prime } \right) = \frac { 5 } { 8 }$\\
(c) find the exact value of $s$
\hfill \mbox{\textit{Edexcel S1 2022 Q6 [11]}}