| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Definitions |
| Type | Listing outcomes and counting |
| Difficulty | Moderate -0.8 This is a straightforward probability question requiring careful reading of a table and basic probability definitions (intersection, union, mutual exclusivity, independence). All parts involve counting outcomes from the given data with no complex calculations or problem-solving insight needed—purely mechanical application of definitions to tabulated information. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space |
| Competiton | ||||||
| 100 m | 200 m | 110 m hurdles | 400 m | Long jump | ||
| \multirow{10}{*}{Athlete} | Abel | ✓ | ✓ | ✓ | ||
| Bernoulli | ✓ | ✓ | ||||
| Cauchy | ✓ | ✓ | ✓ | |||
| Descartes | ✓ | ✓ | ||||
| Einstein | ✓ | ✓ | ||||
| Fermat | ✓ | ✓ | ||||
| Galois | ✓ | ✓ | ||||
| Hardy | ✓ | ✓ | ✓ | |||
| Iwasawa | ✓ | ✓ | ||||
| Jacobi | ✓ | |||||
| Answer | Marks |
|---|---|
| Athletes who can take part in exactly 2 competitions AND 200m: Descartes, Hardy \(\Rightarrow\) Abel takes part in 3, so | |
| \(P(A \cap B) = \frac{2}{10} = \frac{1}{5}\) | B1 |
| Answer | Marks |
|---|---|
| \(P(C \cup D) = \frac{9}{10}\) | B1 |
| Answer | Marks |
|---|---|
| \(C\) and \(D\) are mutually exclusive (no athlete can do both 110m hurdles and long jump) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(B) \times P(D) = \frac{1}{2} \times \frac{2}{5} = \frac{1}{5} \neq \frac{1}{10}\) | M1 A1 | Hence not independent |
# Question 5:
**(i)**
Athletes who can take part in exactly 2 competitions AND 200m: Descartes, Hardy $\Rightarrow$ Abel takes part in 3, so |
$P(A \cap B) = \frac{2}{10} = \frac{1}{5}$ | B1 |
**(ii)**
$C$: Cauchy, Fermat, Jacobi; $D$: Abel, Galois, Hardy, Bernoulli, Einstein, Iwasawa — wait, checking table:
$C \cup D$: Cauchy, Fermat, Jacobi, Abel, Galois, Hardy, Bernoulli, Einstein, Iwasawa = 9
$P(C \cup D) = \frac{9}{10}$ | B1 |
**(iii)**
$C$ and $D$ are mutually exclusive (no athlete can do both 110m hurdles and long jump) | B1 |
**(iv)**
$P(B) = \frac{5}{10} = \frac{1}{2}$, $P(D) = \frac{4}{10} = \frac{2}{5}$
$P(B \cap D) = \frac{1}{10}$ (only Hardy)
$P(B) \times P(D) = \frac{1}{2} \times \frac{2}{5} = \frac{1}{5} \neq \frac{1}{10}$ | M1 A1 | Hence not independent
---5 A school athletics team has 10 members. The table shows which competitions each of the members can take part in.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
& & \multicolumn{5}{|c|}{Competiton} \\
\hline
& & 100 m & 200 m & 110 m hurdles & 400 m & Long jump \\
\hline
\multirow{10}{*}{Athlete} & Abel & ✓ & ✓ & & & ✓ \\
\hline
& Bernoulli & & ✓ & & ✓ & \\
\hline
& Cauchy & ✓ & & ✓ & & ✓ \\
\hline
& Descartes & ✓ & ✓ & & & \\
\hline
& Einstein & & ✓ & & ✓ & \\
\hline
& Fermat & ✓ & & ✓ & & \\
\hline
& Galois & & & & ✓ & ✓ \\
\hline
& Hardy & ✓ & ✓ & & & ✓ \\
\hline
& Iwasawa & & ✓ & & ✓ & \\
\hline
& Jacobi & & & ✓ & & \\
\hline
\end{tabular}
\end{center}
An athlete is selected at random. Events $A , B , C , D$ are defined as follows.\\
A: the athlete can take part in exactly 2 competitions.\\
$B$ : the athlete can take part in the 200 m .\\
$C$ : the athlete can take part in the 110 m hurdles.\\
$D$ : the athlete can take part in the long jump.\\
(i) Write down the value of $\mathrm { P } ( A \cap B )$.\\
(ii) Write down the value of $\mathrm { P } ( C \cup D )$.\\
(iii) Which two of the four events $A , B , C , D$ are mutually exclusive?\\
(iv) Show that events $B$ and $D$ are not independent.
\hfill \mbox{\textit{OCR MEI S1 2006 Q5 [5]}}