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<p>An institution offers classes in three different subjects: Sciences, Arts and Business. The number of students studying different combinations of these subjects are shown in the Venn diagram below.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1779183352541-nwnq2s1dt6f.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzkxODMzNTI1NDEtbnducTJzMWR0NmYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTgwNiwiZXhwIjoxNzkyNzUwMTA2fQ.Vo9NMQmtmDU0YveLsV9qRxXWp16-5FbiWAvxBAJqXqo" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>All students study at least one subject. The number of students who study all three subjects is $x$, and the number of students who study Arts and Business but not Science is $y$. One of the students is chosen at random. $S$, $A$ and $B$ are events defined as follows.</p>
<p style="margin-left: 3rem">•          $S$ is the event that the student studies Science.</p>
<p style="margin-left: 3rem">•          $A$ is the event that the student studies Arts.</p>
<p style="margin-left: 3rem">•          $B$ is the event that the student studies Business.</p>
<p>$A$ and $B$ are not mutually exclusive. Write down a relationship satisfied by $x$ and $y$.<span class="inline-marks">[1]</span></p>
<p>$A$ and $S$ are independent, find an expression for&nbsp;$y$ in terms of $x$.<span class="inline-marks">[3]</span></p>
<p>It is further given that&nbsp;$\mathrm{P}\left( \left. B \right|S \right)=\frac{3}{8}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the values of $x$ and $y$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find $\mathrm{P}\left( \left( S\cap A \right)\cup B' \right)$.<span class="inline-marks">[1]</span></div></li></ol>

An institution offers classes in three different subjects: Sciences, Arts and Business. The number of students studying different combinations of these subjects are shown in the Venn diagram below.

All students study at least one subject. The number of students who study all three subjects is $x$, and the number of students who study Arts and Business but not Science is $y$. One of the students is chosen at random. $S$, $A$ and $B$ are events defined as follows.

•          $S$ is the event that the student studies Science.

•          $A$ is the event that the student studies Arts.

•          $B$ is the event that the student studies Business.

$A$ and $B$ are not mutually exclusive. Write down a relationship satisfied by $x$ and $y$.

$A$ and $S$ are independent, find an expression for $y$ in terms of $x$.

It is further given that $\mathrm{P}\left( \left. B \right|S \right)=\frac{3}{8}$.

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