| Exam Board | Edexcel |
|---|---|
| Module | Paper 3 (Paper 3) |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Definitions |
| Type | Combined event algebra |
| Difficulty | Standard +0.3 This is a straightforward probability question testing standard definitions and formulas. Part (a) requires applying conditional probability and complement rules, (b) tests understanding of independence, (c) involves constructing a Venn diagram with given constraints, and (d) applies De Morgan's laws. All parts use routine techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.03a Mutually exclusive and independent events2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(A' \mid B') = \frac{P(A' \cap B')}{P(B')}\) or \(\frac{0.33}{0.55}\) | M1 | Correct ratio of probabilities formula and at least one correct value |
| \(= \frac{3}{5}\) or 0.6 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| e.g. \(P(A) \times P(B) = \frac{7}{20} \times \frac{9}{20} = \frac{63}{400} \neq P(A \cap B) = 0.13 = \frac{52}{400}\); or \(P(A' \mid B') = 0.6 \neq P(A') = 0.65\) | B1 | Fully correct explanation with correct probabilities and correct comparisons |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Box with \(B\) intersecting \(A\) and \(C\) but \(C\) not intersecting \(A\) | B1 | Accept three intersecting circles with zeros for \(A \cap C\) and \(A \cap B \cap C\); no box is B0 |
| Method for finding \(P(B \cap C)\) | M1 | |
| 0.09 | A1 | |
| 0.13 and their 0.09 in correct places and method for 0.23 | M1 | |
| Fully correct diagram: 0.22, 0.13, 0.23, 0.09, 0.11, 0.22 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(B \cup C)' = 0.22 + 0.22\) or \(1 - [0.56]\) or \(1 - [0.13 + 0.23 + 0.09 + 0.11]\) | M1 | Correct expression, ft their probabilities from Venn diagram |
| \(= 0.44\) | A1 | cao |
# Question 4:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(A' \mid B') = \frac{P(A' \cap B')}{P(B')}$ or $\frac{0.33}{0.55}$ | M1 | Correct ratio of probabilities formula and at least one correct value |
| $= \frac{3}{5}$ or 0.6 | A1 | |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| e.g. $P(A) \times P(B) = \frac{7}{20} \times \frac{9}{20} = \frac{63}{400} \neq P(A \cap B) = 0.13 = \frac{52}{400}$; or $P(A' \mid B') = 0.6 \neq P(A') = 0.65$ | B1 | Fully correct explanation with correct probabilities and correct comparisons |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| Box with $B$ intersecting $A$ and $C$ but $C$ not intersecting $A$ | B1 | Accept three intersecting circles with zeros for $A \cap C$ and $A \cap B \cap C$; no box is B0 |
| Method for finding $P(B \cap C)$ | M1 | |
| 0.09 | A1 | |
| 0.13 and their 0.09 in correct places and method for 0.23 | M1 | |
| Fully correct diagram: 0.22, 0.13, 0.23, 0.09, 0.11, 0.22 | A1 | |
## Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(B \cup C)' = 0.22 + 0.22$ or $1 - [0.56]$ or $1 - [0.13 + 0.23 + 0.09 + 0.11]$ | M1 | Correct expression, ft their probabilities from Venn diagram |
| $= 0.44$ | A1 | cao |\begin{enumerate}
\item Given that
\end{enumerate}
$$\mathrm { P } ( A ) = 0.35 \quad \mathrm { P } ( B ) = 0.45 \quad \text { and } \quad \mathrm { P } ( A \cap B ) = 0.13$$
find\\
(a) $\mathrm { P } \left( A ^ { \prime } \mid B ^ { \prime } \right)$\\
(b) Explain why the events $A$ and $B$ are not independent.
The event $C$ has $\mathrm { P } ( C ) = 0.20$\\
The events $A$ and $C$ are mutually exclusive and the events $B$ and $C$ are statistically independent.\\
(c) Draw a Venn diagram to illustrate the events $A , B$ and $C$, giving the probabilities for each region.\\
(d) Find $\mathrm { P } \left( [ B \cup C ] ^ { \prime } \right)$
\hfill \mbox{\textit{Edexcel Paper 3 Q4 [10]}}