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<p>For events $A$, $B$ and $C$, it is given that&nbsp;$\mathrm{P}(A) = 0.3$, $\mathrm{P}(B) = 0.4$, $\mathrm{P}(C) = 0.42$, $\mathrm{P}(A' \cap B \cap C) = x$ and $\mathrm{P}(A \cap B \cap C) = 0.02$. It is also given that events $A$ and $B$ are independent, and that events $A$ and $C$ are independent.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find $\mathrm{P}(A \cap B)$ and $\mathrm{P}(A \cap C)$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Draw a Venn diagram to represent this situation, showing the probability in each of the eight regions, in terms of $x$ where necessary.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the greatest and least possible values of $\mathrm{P}(A' \cap B' \cap C)$.</span><span class="inline-marks">[2]</span></div></li></ol>

For events $A$, $B$ and $C$, it is given that $\mathrm{P}(A) = 0.3$, $\mathrm{P}(B) = 0.4$, $\mathrm{P}(C) = 0.42$, $\mathrm{P}(A' \cap B \cap C) = x$ and $\mathrm{P}(A \cap B \cap C) = 0.02$. It is also given that events $A$ and $B$ are independent, and that events $A$ and $C$ are independent.

Find $\mathrm{P}(A \cap B)$ and $\mathrm{P}(A \cap C)$.

Draw a Venn diagram to represent this situation, showing the probability in each of the eight regions, in terms of $x$ where necessary.

Find the greatest and least possible values of $\mathrm{P}(A' \cap B' \cap C)$.

Probability from Venn Diagrams

2025 TMJC P2 Q6

Probability

<p>A group of $100$ students are asked if they are student leaders, in a sports CCA, or studying in a science faculty. The number of students who are student leaders is $25$, the number of students who are in a sports CCA is $30$ and the number of students studying in a science faculty is $40$. There are $15$ student leaders who are in a sports CCA. The number of students who are student leaders, in a sports CCA and studying in a science faculty is $x$. The number of students who are in a sports CCA and studying in a science faculty but not a student leader is $y$.</p>
<p>One of the students is chosen at random.</p>
<p style="margin-left: 4rem">$A$ is the event that the student is a student leader.<br>$B$ is the event that the student is in a sports CCA.<br>$C$ is the event that the student is studying in a science faculty.</p>
<p>It is given that&nbsp;$A$ and $C$ are independent.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Complete the Venn diagram below to represent all the above information. You are allowed to give expressions in terms of $x$ and $y$.<span class="inline-marks">[3]</span></div></li></ol>
<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/1768641136372-4cabuxtxvzw.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg2NDExMzYzNzItNGNhYnV4dHh2encucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTY4NCwiZXhwIjoxNzkyNzQ5OTg0fQ.2OXmuerZxJpR91NbOxW2G0y3qJQcZcQrBRu5WQDEZ-M" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>It is further given that&nbsp;$B$ and $C$ are independent.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find $y$ in terms of $x$. Hence, find the greatest and least possible values of $y$.<span class="inline-marks">[4]</span></div></li></ol>

A group of $100$ students are asked if they are student leaders, in a sports CCA, or studying in a science faculty. The number of students who are student leaders is $25$, the number of students who are in a sports CCA is $30$ and the number of students studying in a science faculty is $40$. There are $15$ student leaders who are in a sports CCA. The number of students who are student leaders, in a sports CCA and studying in a science faculty is $x$. The number of students who are in a sports CCA and studying in a science faculty but not a student leader is $y$.

$A$ is the event that the student is a student leader.
$B$ is the event that the student is in a sports CCA.
$C$ is the event that the student is studying in a science faculty.

It is given that $A$ and $C$ are independent.

Complete the Venn diagram below to represent all the above information. You are allowed to give expressions in terms of $x$ and $y$.

It is further given that $B$ and $C$ are independent.

Find $y$ in terms of $x$. Hence, find the greatest and least possible values of $y$.

Probability Bounds and Inequalities

2025 CJC P2 Q6

<p>For events $A$ , $B$ and $C$ it is given that&nbsp;$\mathrm{P}\left( A \right)\mathrm{=P}\left( A\cap B \right)=\frac{1}{5}$ and $\mathrm{P}\left( B \right)\mathrm{=2P}\left( A \right)$. It is also given that&nbsp;$\mathrm{P}\left( B\cap C \right)=\frac{3}{10}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the greatest and least possible values of $\mathrm{P}\left( A\cap C \right)$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Draw a Venn diagram showing all $3$ events representing the case where the value of $\mathrm{P}\left( A\cap C \right)$ is the greatest.<span class="inline-marks">[1]</span></div></li></ol>
<p>It is now known that&nbsp;$\mathrm{P}\left( A'\cap B'\cap C' \right)=\frac{1}{12}$.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">If&nbsp;$A$ and $C$ are independent events, calculate the value of $\mathrm{P}\left( A\cap C \right)$.<span class="inline-marks">[2]</span></div></li></ol>

For events $A$ , $B$ and $C$ it is given that $\mathrm{P}\left( A \right)\mathrm{=P}\left( A\cap B \right)=\frac{1}{5}$ and $\mathrm{P}\left( B \right)\mathrm{=2P}\left( A \right)$. It is also given that $\mathrm{P}\left( B\cap C \right)=\frac{3}{10}$.

It is now known that $\mathrm{P}\left( A'\cap B'\cap C' \right)=\frac{1}{12}$.

Independent Events

Complementary Events

Conditional Probability

Mutually Exclusive Events

2023 EJC P2 Q6

<p>The events $A$ and $B$ are such that&nbsp;$\mathrm{P}\left( A \right)=a$ and $\mathrm{P}\left( B \right)=b$. $A$ and $B$ are mutually exclusive events.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find an expression for&nbsp;$\mathrm{P}\left( {A}'\cap {B}' \right)$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Draw a Venn diagram to illustrate the case where&nbsp;${A}'$ and ${B}'$ are also mutually exclusive events. Hence, state an equation relating $a$ and $b$.<span class="inline-marks">[2]</span></div></li></ol>
<p>For the rest of the question, ${A}'$ and ${B}'$ are not mutually exclusive events. The event $C$ is such that&nbsp;$\mathrm{P}\left( C \right)=c$. $A$ and $C$ are independent events. It is also given that&nbsp;$a=0.2$, $b=0.3$ and $\mathrm{P}\left( B\cap C \right)=0.14$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find an expression for&nbsp;$\mathrm{P}\left( A\cup B\cup C \right)$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find an expression for&nbsp;$\mathrm{P}\left( {A}'\cap {B}'\cap C \right)$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Use your answers to parts <strong>(iii)</strong> and <strong>(iv)</strong> to find the range of values of $c$.<span class="inline-marks">[2]</span></div></li></ol>

The events $A$ and $B$ are such that $\mathrm{P}\left( A \right)=a$ and $\mathrm{P}\left( B \right)=b$. $A$ and $B$ are mutually exclusive events.

For the rest of the question, ${A}'$ and ${B}'$ are not mutually exclusive events. The event $C$ is such that $\mathrm{P}\left( C \right)=c$. $A$ and $C$ are independent events. It is also given that $a=0.2$, $b=0.3$ and $\mathrm{P}\left( B\cap C \right)=0.14$.

Addition Rule for Overlapping Events

2023 JPJC P2 Q8

<p>The events $A$, $B$ and $C$ are such that&nbsp;$P(A)=a$, $\mathrm{P}\left( B \right)=b$ and $\mathrm{P}\left( C \right)=c$. $A$ and $B$ are independent events. $A$ and $C$ are mutually exclusive events.</p>
<ol class="i-list"><li data-list-item-id="item-1759647727529-tcxhlcglh" data-list-item-path="item-1759647727529-tcxhlcglh"><div class="list-item-paragraph">Find an expression for&nbsp;$\mathrm{P}\left( A'\cap B' \right)$ and hence prove that&nbsp;$A'$ and $B'$ are independent events.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759647729328-v2rui4rfl" data-list-item-path="item-1759647729328-v2rui4rfl"><div class="list-item-paragraph">Find an expression for&nbsp;$\mathrm{P}\left( A'\cap C' \right)$. Draw a Venn diagram to illustrate to case when&nbsp;$A'$ and $C'$ are also mutually exclusive events. (You should not show event $B$on your diagram.)<span class="inline-marks">[2]</span></div></li></ol>
<p>You are now given that&nbsp;$A'$ and $C'$ are not mutually exclusive, $P(A)=\frac{2}{5}$, $\mathrm{P}\left( B\cap C \right)=\frac{1}{5}$ and $\mathrm{P}\left( A'\cap B'\cap C' \right)=\frac{1}{10}$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1759647750403-v6wixze04" data-list-item-path="item-1759647750403-v6wixze04"><div class="list-item-paragraph">Find exactly the maximum and minimum possible values of $\mathrm{P}\left( A\cap B \right)$.<span class="inline-marks">[4]</span></div></li></ol>

The events $A$, $B$ and $C$ are such that $P(A)=a$, $\mathrm{P}\left( B \right)=b$ and $\mathrm{P}\left( C \right)=c$. $A$ and $B$ are independent events. $A$ and $C$ are mutually exclusive events.

You are now given that $A'$ and $C'$ are not mutually exclusive, $P(A)=\frac{2}{5}$, $\mathrm{P}\left( B\cap C \right)=\frac{1}{5}$ and $\mathrm{P}\left( A'\cap B'\cap C' \right)=\frac{1}{10}$.

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