| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2002 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Independent Events |
| Type | Test independence using definition |
| Difficulty | Moderate -0.8 This is a straightforward probability question testing basic set operations and independence definition. Parts (a)-(c) use standard Venn diagram or complement rules with given probabilities. Part (d) tests the definition P(A∩B) = P(A)×P(B), requiring only arithmetic comparison. All techniques are routine S1 content with no problem-solving insight needed. |
| Spec | 2.03a Mutually exclusive and independent events |
| Answer | Marks | Guidance |
|---|---|---|
| Venn diagram | M1 | 0.3, 0.2, 0.1 |
| \(P(\text{does not win either}) = 0.4\) | A1 (3) |
| Answer | Marks |
|---|---|
| \(P(\text{wins exactly one}) = 0.3 + 0.1 = 0.4\) | M1 |
| A1/(2) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(B_1 | B_1') = \frac{P(B_2 \cap B_1')}{P(B_1')} = \frac{0.1}{0.5} = 0.2\) | M1 |
| A1 (2) |
| Answer | Marks | Guidance |
|---|---|---|
| For independence: \(P(B_1 \cap B_2) = P(B_1) \times P(B_2)\) | M1 | |
| \(P(B_1 \cap B_2) = 0.2\); \(P(B_1) \times P(B_2) = 0.15\) | 0.2; 0.15 | A1 |
| LHS \(\neq\) RHS \(\Rightarrow\) events not independent | A1 (3) | |
| NB: Accept alternate correct solutions |
## Part (a)
Venn diagram | M1 | 0.3, 0.2, 0.1
$P(\text{does not win either}) = 0.4$ | A1 (3) |
## Part (b)
$P(\text{wins exactly one}) = 0.3 + 0.1 = 0.4$ | M1 |
| A1/(2) |
## Part (c)
$P(B_1 | B_1') = \frac{P(B_2 \cap B_1')}{P(B_1')} = \frac{0.1}{0.5} = 0.2$ | M1 | Use of conditional probability
| A1 (2) |
## Part (d)
For independence: $P(B_1 \cap B_2) = P(B_1) \times P(B_2)$ | M1 |
$P(B_1 \cap B_2) = 0.2$; $P(B_1) \times P(B_2) = 0.15$ | 0.2; 0.15 | A1
LHS $\neq$ RHS $\Rightarrow$ events not independent | | A1 (3)
NB: Accept alternate correct solutions | |
---A contractor bids for two building projects. He estimates that the probability of winning the first project is 0.5, the probability of winning the second is 0.3 and the probability of winning both projects is 0.2.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that he does not win either project. [3]
\item Find the probability that he wins exactly one project. [2]
\item Given that he does not win the first project, find the probability that he wins the second. [2]
\item By calculation, determine whether or not winning the first contract and winning the second contract are independent events. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2002 Q4 [10]}}