| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2020 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Independent Events |
| Type | Independence in contingency tables |
| Difficulty | Moderate -0.8 This is a straightforward contingency table question requiring basic probability calculations: reading values from a table, computing simple probabilities, and checking independence using P(A∩B) = P(A)×P(B). All values are given directly in the table with no algebraic manipulation or conceptual insight needed—purely mechanical application of standard formulas. |
| Spec | 2.03a Mutually exclusive and independent events2.03c Conditional probability: using diagrams/tables |
| \cline { 2 - 4 } \multicolumn{1}{c|}{} | Soccer | Hockey | Total |
| Amos | 54 | 32 | 86 |
| Benn | 84 | 72 | 156 |
| Canton | 22 | 56 | 78 |
| Devar | 120 | 60 | 180 |
| Total | 280 | 220 | 500 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{56}{500}\) or \(\frac{14}{125}\) or \(0.112\) | B1 | |
| Total | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(D\ | S) = \dfrac{P(D \cap S)}{P(S)} = \dfrac{120}{280}\) | M1 |
| \(\dfrac{120}{280}\) or \(\dfrac{3}{7}\) | A1 | |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(\text{hockey}) = \frac{220}{500} = 0.44\); \(P(\text{Amos or Benn}) = \frac{242}{500} = 0.484\); \(P(\text{hockey} \cap (A \cup B)) = \frac{104}{500} = 0.208\); \(P(H) \times P(A \cup B) = P(H \cap (A \cup B))\) if independent | M1 | Method for testing independence |
| \(\frac{220}{500} \times \frac{242}{500} = \frac{1331}{6250} \neq 0.208\), so not independent | A1 |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{56}{500}$ or $\frac{14}{125}$ or $0.112$ | B1 | |
| **Total** | **1** | |
---
## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(D\|S) = \dfrac{P(D \cap S)}{P(S)} = \dfrac{120}{280}$ | M1 | |
| $\dfrac{120}{280}$ or $\dfrac{3}{7}$ | A1 | |
| **Total** | **2** | |
## Question 2(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(\text{hockey}) = \frac{220}{500} = 0.44$; $P(\text{Amos or Benn}) = \frac{242}{500} = 0.484$; $P(\text{hockey} \cap (A \cup B)) = \frac{104}{500} = 0.208$; $P(H) \times P(A \cup B) = P(H \cap (A \cup B))$ if independent | M1 | Method for testing independence |
| $\frac{220}{500} \times \frac{242}{500} = \frac{1331}{6250} \neq 0.208$, so not independent | A1 | |
---2 A total of 500 students were asked which one of four colleges they attended and whether they preferred soccer or hockey. The numbers of students in each category are shown in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | }
\cline { 2 - 4 }
\multicolumn{1}{c|}{} & Soccer & Hockey & Total \\
\hline
Amos & 54 & 32 & 86 \\
\hline
Benn & 84 & 72 & 156 \\
\hline
Canton & 22 & 56 & 78 \\
\hline
Devar & 120 & 60 & 180 \\
\hline
Total & 280 & 220 & 500 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Find the probability that a randomly chosen student is at Canton college and prefers hockey.
\item Find the probability that a randomly chosen student is at Devar college given that he prefers soccer.
\item One of the students is chosen at random. Determine whether the events 'the student prefers hockey' and 'the student is at Amos college or Benn college' are independent, justifying your answer.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2020 Q2 [5]}}