SPS SPS SM Statistics 2023 January — Question 7 11 marks

Exam BoardSPS
ModuleSPS SM Statistics (SPS SM Statistics)
Year2023
SessionJanuary
Marks11
TopicPrinciple of Inclusion/Exclusion
TypeFinding Unknown Probabilities in Venn Diagrams
DifficultyStandard +0.3 This is a standard Venn diagram problem requiring systematic application of probability rules (sum to 1, conditional probability, independence). While it has multiple parts, each step follows directly from basic definitions without requiring novel insight or complex problem-solving—slightly easier than average A-level.
Spec2.03a 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
7. A large college produces three magazines.
One magazine is about green issues, one is about equality and one is about sports. A student at the college is selected at random and the events \(G , E\) and \(S\) are defined as follows \(G\) is the event that the student reads the magazine about green issues \(E\) is the event that the student reads the magazine about equality \(S\) is the event that the student reads the magazine about sports The Venn diagram, where \(p , q , r\) and \(t\) are probabilities, gives the probability for each subset. \includegraphics[max width=\textwidth, alt={}, center]{f03113c4-039e-4ead-9588-b4b83fb7eea9-12_533_903_790_548}
  1. Find the proportion of students in the college who read exactly one of these magazines. No students read all three magazines and \(\mathrm { P } ( G ) = 0.25\)
  2. Find
    1. the value of \(p\)
    2. the value of \(q\) Given that \(\mathrm { P } ( S \mid E ) = \frac { 5 } { 12 }\)
  3. find
    1. the value of \(r\)
    2. the value of \(t\)
  4. Determine whether or not the events ( \(S \cap E ^ { \prime }\) ) and \(G\) are independent. Show your working clearly. END OF TEST
7.

A large college produces three magazines.\\
One magazine is about green issues, one is about equality and one is about sports. A student at the college is selected at random and the events $G , E$ and $S$ are defined as follows\\
$G$ is the event that the student reads the magazine about green issues $E$ is the event that the student reads the magazine about equality $S$ is the event that the student reads the magazine about sports

The Venn diagram, where $p , q , r$ and $t$ are probabilities, gives the probability for each subset.\\
\includegraphics[max width=\textwidth, alt={}, center]{f03113c4-039e-4ead-9588-b4b83fb7eea9-12_533_903_790_548}
\begin{enumerate}[label=(\alph*)]
\item Find the proportion of students in the college who read exactly one of these magazines.

No students read all three magazines and $\mathrm { P } ( G ) = 0.25$
\item Find
\begin{enumerate}[label=(\roman*)]
\item the value of $p$
\item the value of $q$

Given that $\mathrm { P } ( S \mid E ) = \frac { 5 } { 12 }$
\end{enumerate}\item find
\begin{enumerate}[label=(\roman*)]
\item the value of $r$
\item the value of $t$
\end{enumerate}\item Determine whether or not the events ( $S \cap E ^ { \prime }$ ) and $G$ are independent. Show your working clearly.

END OF TEST
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM Statistics 2023 Q7 [11]}}
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