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E Math (IGCSE)

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H2 Math

IB AA HL Math

IB AA SL Math

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IB AI SL Math

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N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>Let&nbsp;$A$ and $B$ be independent events, where&nbsp;$P(A) = 0.6$ and $P(B) = x$.</p>
<ol class="abc-list"><li data-list-item-id="ibsl-q033-part-1-003" data-list-item-path="ibsl-q033-part-1-003"><div class="list-item-paragraph">Write down an expression for&nbsp;$P(A ∩ B)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="ibsl-q033-part-2-004" data-list-item-path="ibsl-q033-part-2-004"><div class="list-item-paragraph"><p>Given that&nbsp;$P(A ∪ B) = 0.8$,</p><span class="inline-marks">[4]</span></div><ol class="i-list"><li data-list-item-id="ibsl-q033-part-2-1-005" data-list-item-path="ibsl-q033-part-2-004/ibsl-q033-part-2-1-005"><div class="list-item-paragraph"><p>find $x$;</p></div></li><li data-list-item-id="ibsl-q033-part-2-2-006" data-list-item-path="ibsl-q033-part-2-004/ibsl-q033-part-2-2-006"><div class="list-item-paragraph">find $P(A ∩ B)$.</div></li></ol></li><li data-list-item-id="ibsl-q033-part-3-007" data-list-item-path="ibsl-q033-part-3-007"><div class="list-item-paragraph"><p>Hence, explain why $A$ and $B$ are not mutually exclusive.</p><span class="inline-marks">[1]</span></div></li></ol>

Let $A$ and $B$ be independent events, where $P(A) = 0.6$ and $P(B) = x$.

Write down an expression for $P(A ∩ B)$.

Set Theory and Probability — Combined Problems

Independent Events

Mutually Exclusive Events

Use independence to solve an unknown event probability and distinguish mutual exclusivity.

Probability

<p>The following Venn diagram shows a sample space $U$ and events $A$ and $B$. It is known that&nbsp;$n(U)=36$, $n(A)=11$, $n(B)=6$, and $n((A\cup B)')=21$.</p>
<div class="graph-block" data-graph-config="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" data-width="520" data-height="260" style="width:520px;height:260px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="abc-list"><li data-list-item-id="ibsl-q101-part-1-010" data-list-item-path="ibsl-q101-part-1-010"><div class="list-item-paragraph"><p>On the diagram, shade the region $(A ∪ B)′$.</p></div></li><li data-list-item-id="ibsl-q101-part-2-011" data-list-item-path="ibsl-q101-part-2-011"><div class="list-item-paragraph">Find</div><ol class="i-list"><li data-list-item-id="ibsl-q101-part-2-1-012" data-list-item-path="ibsl-q101-part-2-011/ibsl-q101-part-2-1-012"><div class="list-item-paragraph"><p>$n(A ∩ B)$;</p></div></li><li data-list-item-id="ibsl-q101-part-2-2-013" data-list-item-path="ibsl-q101-part-2-011/ibsl-q101-part-2-2-013"><div class="list-item-paragraph">$P(A ∩ B)$.</div></li></ol></li><li data-list-item-id="ibsl-q101-part-3-014" data-list-item-path="ibsl-q101-part-3-014"><div class="list-item-paragraph"><p>Explain why events $A$ and $B$ are not mutually exclusive.</p></div></li></ol>
<p><span class="inline-marks">[4]</span></p>

The following Venn diagram shows a sample space $U$ and events $A$ and $B$. It is known that $n(U)=36$, $n(A)=11$, $n(B)=6$, and $n((A\cup B)')=21$.

Shade a Venn complement, calculate an intersection and explain non-mutual-exclusivity.

<p>A bag contains numbered balls from $2$ to $31$. A ball is picked at random from the bag. Find the probability that the number on the ball is</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">either a perfect square or a factor of $30$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">either a prime number or multiple of $4$.</span></div></li></ol>

A bag contains numbered balls from $2$ to $31$. A ball is picked at random from the bag. Find the probability that the number on the ball is

either a perfect square or a factor of $30$,

either a prime number or multiple of $4$.

Diagnostic Assessment: Mutually Exclusive vs. Independent Events

<p>Ariel and Ben are taking part in a swimming race. There can only be one winner. The probability of Ariel winning the race is $\frac{1}{3}$. The probability of Ben winning the race is $\frac{1}{5}$. Find the probability that</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">either Ariel or Ben wins,</span></div></li><li><div class="list-item-content"><span class="list-item-text">neither Ariel nor Ben wins.</span></div></li></ol>

Ariel and Ben are taking part in a swimming race. There can only be one winner. The probability of Ariel winning the race is $\frac{1}{3}$. The probability of Ben winning the race is $\frac{1}{5}$. Find the probability that

Complementary Events

Mutually Exclusive Events: Axioms of Probability and Summation

<p>For events $A$ and $B$, it is given that&nbsp;$P\left( A \right)=\frac{1}{8}$, $P\left( A\cup B \right)=\frac{9}{16}$ and $P\left( A\cap {B}' \right)=\frac{1}{16}$.</p>
<ol class="i-list"><li data-list-item-id="item-1760781172283-qq04lwcra" data-list-item-path="item-1760781172283-qq04lwcra"><div class="list-item-paragraph">Find $P\left( B \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760781174061-kzrl5fl7k" data-list-item-path="item-1760781174061-kzrl5fl7k"><div class="list-item-paragraph">Determine, with a reason, whether the events $A$ and $B$ are independent.<span class="inline-marks">[2]</span></div></li></ol>
<p>For a third event $C$, it is given that&nbsp;$P\left( C \right)=\frac{3}{8}$, $P\left( A\cap B\cap C \right)=\frac{1}{64}$ and $C$ is independent of $A$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760781194927-07nr12r5q" data-list-item-path="item-1760781194927-07nr12r5q"><div class="list-item-paragraph">Find $P\left( {A}'\cap C \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760781195819-twm664oux" data-list-item-path="item-1760781195819-twm664oux"><div class="list-item-paragraph">Find the range of values for&nbsp;$P\left( A'\cap {B}'\cap {C}' \right)$.<span class="inline-marks">[3]</span></div></li></ol>

For events $A$ and $B$, it is given that $P\left( A \right)=\frac{1}{8}$, $P\left( A\cup B \right)=\frac{9}{16}$ and $P\left( A\cap {B}' \right)=\frac{1}{16}$.

For a third event $C$, it is given that $P\left( C \right)=\frac{3}{8}$, $P\left( A\cap B\cap C \right)=\frac{1}{64}$ and $C$ is independent of $A$.

Conditional Probability

Probability Bounds and Inequalities

Addition Rule for Overlapping Events

2024 TMJC P2 Q5

<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 independent events.</p>
<ol class="abc-list"><li data-list-item-id="item-1760781012593-etiznm953" data-list-item-path="item-1760781012593-etiznm953"><div class="list-item-paragraph">Find an expression for&nbsp;$\mathrm{P}\left( {A}'\cap {B}' \right)$ in terms of $a$ and $b$, and hence prove that&nbsp;${A}'$ and ${B}'$ are independent events.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is given that&nbsp;$\mathrm{P}\left( A\mid {B}' \right)=0.85$ and ${\mathrm{P}\left( {{B}'} \right)=0.8}$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760781026636-y4nwlf8ee" data-list-item-path="item-1760781026636-y4nwlf8ee"><div class="list-item-paragraph">Find $\mathrm{P}\left( A\cap {B}' \right)$.<span class="inline-marks">[2]</span></div></li></ol>
<p>For a third event $C$, it is given that&nbsp;$A$ and $C$ are mutually exclusive and $\mathrm{P}\left( {A}'\cap {C}' \right)=0.52$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-1760781041671-n28p3vsy0" data-list-item-path="item-1760781041671-n28p3vsy0"><div class="list-item-paragraph">Find $\mathrm{P}\left( C \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760781044055-x873mmx49" data-list-item-path="item-1760781044055-x873mmx49"><div class="list-item-paragraph">Hence find the set of possible values of $\mathrm{P}\left( {A}'\cap {B}'\cap {C}' \right)$.<span class="inline-marks">[3]</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 independent events.

It is given that $\mathrm{P}\left( A\mid {B}' \right)=0.85$ and ${\mathrm{P}\left( {{B}'} \right)=0.8}$.

For a third event $C$, it is given that $A$ and $C$ are mutually exclusive and $\mathrm{P}\left( {A}'\cap {C}' \right)=0.52$.

2024 ACJC P2 Q6

<p>For events $A$, $B$ and $C$, it is given that&nbsp;$\mathrm{P}\left( A \right)=0.4$, $\mathrm{P}\left( C \right)=0.3$ and $\mathrm{P}\left( B\cap C \right)=0.1$. It is also given that events $A$ and $B$ are mutually exclusive, and that events $A$ and $C$ are independent.</p>
<ol class="i-list"><li data-list-item-id="item-1760434547091-w2qaob5f6" data-list-item-path="item-1760434547091-w2qaob5f6"><div class="list-item-paragraph">Find $\mathrm{P}\left( A\cap B \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760434547998-2m87gz01q" data-list-item-path="item-1760434547998-2m87gz01q"><div class="list-item-paragraph">Find $\mathrm{P}\left( \left. A \right|C \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760434548854-avy5o3p8z" data-list-item-path="item-1760434548854-avy5o3p8z"><div class="list-item-paragraph">Given also that events $B$ and $C$ are independent, find $\mathrm{P}\left( A'\cap B'\cap C' \right)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760434549702-t36ytqf4y" data-list-item-path="item-1760434549702-t36ytqf4y"><div class="list-item-paragraph">Given instead that events $B$ and $C$ are not independent, find the greatest and least possible values of $\mathrm{P}\left( B \right)$.<span class="inline-marks">[2]</span></div></li></ol>

For events $A$, $B$ and $C$, it is given that $\mathrm{P}\left( A \right)=0.4$, $\mathrm{P}\left( C \right)=0.3$ and $\mathrm{P}\left( B\cap C \right)=0.1$. It is also given that events $A$ and $B$ are mutually exclusive, and that events $A$ and $C$ are independent.

2023 MI P2 Q8

<p>For two independent events $A$ and $B$, it is given that&nbsp;$\mathrm{P}\left( A \right)=a$ and $\mathrm{P}\left( B \right)=b$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760431524339-ghq9dx4cn" data-list-item-path="item-1760431524339-ghq9dx4cn"><div class="list-item-paragraph">Show that&nbsp;$A'$ and $B'$ are independent events.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is given that&nbsp;$\mathrm{P}\left( A\cup B \right)=0.7$ and $b=0.5$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760431537994-knj9ko8gc" data-list-item-path="item-1760431537994-knj9ko8gc"><div class="list-item-paragraph">Show that&nbsp;$a=0.4$.<span class="inline-marks">[2]</span></div></li></ol>
<p>For a third event $C$, it is given that&nbsp;$\mathrm{P}\left( C \right)=0.3$ and events $A$ and $C$ are independent.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-1760431561438-xcawbowjj" data-list-item-path="item-1760431561438-xcawbowjj"><div class="list-item-paragraph">If events $B$ and $C$ are mutually exclusive, find $\mathrm{P}\left( A'\cap B'\cap C' \right)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760431562521-qj5mj53yr" data-list-item-path="item-1760431562521-qj5mj53yr"><div class="list-item-paragraph">If events $B$ and $C$ are not mutually exclusive, find the greatest possible value of $\mathrm{P}\left( A'\cap B'\cap C' \right)$, showing your working clearly.<span class="inline-marks">[3]</span></div></li></ol>

For two independent events $A$ and $B$, it is given that $\mathrm{P}\left( A \right)=a$ and $\mathrm{P}\left( B \right)=b$.

It is given that $\mathrm{P}\left( A\cup B \right)=0.7$ and $b=0.5$.

For a third event $C$, it is given that $\mathrm{P}\left( C \right)=0.3$ and events $A$ and $C$ are independent.

2023 RI P2 Q7

<p>The events $A$, $B$ and $C$ are such that&nbsp;$\mathrm{P}\left( A \right)=0.1$, $\mathrm{P}\left( B \right)=0.2$ and $\mathrm{P}\left( C \right)=c$. $A$ and $B$ are independent events. $A$ and $C$ are mutually exclusive events.</p>
<ol class="abc-list"><li data-list-item-id="item-1760431416676-nxza23cfd" data-list-item-path="item-1760431416676-nxza23cfd"><div class="list-item-paragraph">State the values of $\mathrm{P}\left( \left. A \right|B \right)$ and $\mathrm{P}\left( \left. A \right|C \right)$.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is given that&nbsp;$\mathrm{P}\left( A\cup B \right)=\mathrm{P}\left( A'\cap C' \right)$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760431426253-xf6s1zg99" data-list-item-path="item-1760431426253-xf6s1zg99"><div class="list-item-paragraph">Find the value of $c$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760431426921-0mwvjn2nu" data-list-item-path="item-1760431426921-0mwvjn2nu"><div class="list-item-paragraph">Find the maximum and minimum possible values of $\mathrm{P}\left( A'\cap B'\cap C' \right)$.<span class="inline-marks">[4]</span></div></li></ol>

The events $A$, $B$ and $C$ are such that $\mathrm{P}\left( A \right)=0.1$, $\mathrm{P}\left( B \right)=0.2$ and $\mathrm{P}\left( C \right)=c$. $A$ and $B$ are independent events. $A$ and $C$ are mutually exclusive events.

It is given that $\mathrm{P}\left( A\cup B \right)=\mathrm{P}\left( A'\cap C' \right)$.

2023 NJC P2 Q8

<p>For events $A$ and $B$, it is given that&nbsp;$\mathrm{P}\left( A'\cup B' \right)=0.82$ and $\mathrm{P}\left( A|B \right)=0.4$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760431034184-t73jio1gy" data-list-item-path="item-1760431034184-t73jio1gy"><div class="list-item-paragraph">Find $\mathrm{P}\left( B \right)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760431035864-j71cytbo5" data-list-item-path="item-1760431035864-j71cytbo5"><div class="list-item-paragraph">Hence find an inequality for&nbsp;$\mathrm{P}\left( A\cap B' \right)$.<span class="inline-marks">[2]</span></div></li></ol>
<p>A third event $C$ is such that&nbsp;$\mathrm{P}\left( C \right)=0.1$. $A$ and $C$ are mutually exclusive. $B$ and $C$ are independent.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-1760431057115-tq0pslf3j" data-list-item-path="item-1760431057115-tq0pslf3j"><div class="list-item-paragraph">Find $\mathrm{P}\left( A'\cap B\cap C' \right)$.<span class="inline-marks">[3]</span></div></li></ol>

For events $A$ and $B$, it is given that $\mathrm{P}\left( A'\cup B' \right)=0.82$ and $\mathrm{P}\left( A|B \right)=0.4$.

A third event $C$ is such that $\mathrm{P}\left( C \right)=0.1$. $A$ and $C$ are mutually exclusive. $B$ and $C$ are independent.

2023 YIJC P2 Q6

<p>For events $A$, $B$ and $C$ it is given that $\mathrm{P}\left( A \right)=0.45$, $\mathrm{P}\left( B \right)=0.4$, $\mathrm{P}\left( C \right)=0.3$ and $P\left( A\cap B\cap C \right)=0.1$. It is also given that events $A$ and $B$ are independent, and that events $A$ and $C$ are independent.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find $\mathrm{P}\left( B|A \right)$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Given also that event $B$ and $C$ are independent, find $\mathrm{P}\left( A'\cap B'\cap C' \right)$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given instead that events $B$ and $C$ are not independent, find the greatest and least possible values of $\mathrm{P}\left( A'\cap B'\cap C' \right)$.</div><p style="text-align: right;">[4]</p></li></ol>

For events $A$, $B$ and $C$ it is given that $\mathrm{P}\left( A \right)=0.45$, $\mathrm{P}\left( B \right)=0.4$, $\mathrm{P}\left( C \right)=0.3$ and $P\left( A\cap B\cap C \right)=0.1$. It is also given that events $A$ and $B$ are independent, and that events $A$ and $C$ are independent.

N2015 P2 Q9

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzkxODMzNTI1NDEtbnducTJzMWR0NmYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTY0MSwiZXhwIjoxNzkyNzQ5OTQxfQ.DliLPT8GhsNBXLJ0Lq1fLFlOUYI8KkhQ1s1hU3NAgcA" 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 $\mathrm{P}\left( \left. B \right|S \right)=\frac{3}{8}$.

Conditional Probability from Venn Diagrams

2022 RVHS P2 Q8

<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 $\mathrm{P}\left( A'\cap B'\cap C' \right)=\frac{1}{12}$.

Probability from Venn Diagrams

2023 EJC P2 Q6

<p>For events $A$ and $B$, it is given that&nbsp;$\mathrm{P}\left( B' \right)=0.51$, $\mathrm{P}\left( A'\cap B' \right)=0.15$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find $\mathrm{P}\left( A\cap B' \right)$.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is also given that&nbsp;$\mathrm{P}\left( B\left| \,A \right. \right)=\frac{11}{29}$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find $\mathrm{P}\left( A \right)$.<span class="inline-marks">[3]</span></div></li></ol>
<p>For a third event $C$, it is given that&nbsp;$A$ and $C$ are independent of each other, and $B$ and $C$ are mutually exclusive.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Find the maximum value of $\mathrm{P}\left( C \right)$.<span class="inline-marks">[3]</span></div></li></ol>

For events $A$ and $B$, it is given that $\mathrm{P}\left( B' \right)=0.51$, $\mathrm{P}\left( A'\cap B' \right)=0.15$.

It is also given that $\mathrm{P}\left( B\left| \,A \right. \right)=\frac{11}{29}$.

For a third event $C$, it is given that $A$ and $C$ are independent of each other, and $B$ and $C$ are mutually exclusive.

2023 TMJC P2 Q5

<p>For independent events $A$ and $B$, it is given that&nbsp;$\mathrm{P}(A\cap B)=0.05$ and $\mathrm{P}\left( A\cup B \right)=0.55$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find $\mathrm{P}\left( A \right)$ and $\mathrm{P}\left( B \right)$ if&nbsp;$\mathrm{P}\left( A \right)&lt;\mathrm{P}\left( B \right)$.<span class="inline-marks">[4]</span></div></li></ol>
<p>For a third event $C$, $\mathrm{P}\left( C \right)=0.5$ and $\mathrm{P}\left( A'\cup B'|C \right)=0.95$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find $\mathrm{P}\left( A\cap B\cap C \right)$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">State, with reason, whether the events $A$  and $C$ are mutually exclusive.<span class="inline-marks">[1]</span></div></li></ol>

For independent events $A$ and $B$, it is given that $\mathrm{P}(A\cap B)=0.05$ and $\mathrm{P}\left( A\cup B \right)=0.55$.

For a third event $C$, $\mathrm{P}\left( C \right)=0.5$ and $\mathrm{P}\left( A'\cup B'|C \right)=0.95$.

Finding Unknowns in Probability Models

2015 DHS P2 Q9

<p>This question is about arrangements of all eight letters in the word IMMUNITY.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the number of different arrangements of the eight letters that can be made is $10080$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the number of different arrangements that can be made with no two vowels next to each other.</div><p style="text-align: right;">[3]</p></li></ol>
<p>One of the $10080$ arrangements in part <strong>(i)</strong> is randomly chosen. Let $A$ denote the event that the two I’s are next to each other and let $B$ denote the event that the two M’s are next to each other.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Determine, with a reason, whether $A$ and $B$ are</div><ol class="abc-list"><li><div class="list-item-paragraph">mutually exclusive,</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">independent.</div><p style="text-align: right;">[3]</p></li></ol></li><li><div class="list-item-paragraph">Find the probability that the chosen arrangement contains no two adjacent letters that are the same.</div><p style="text-align: right;">[4]</p></li></ol>

This question is about arrangements of all eight letters in the word IMMUNITY.

One of the $10080$ arrangements in part **(i)** is randomly chosen. Let $A$ denote the event that the two I’s are next to each other and let $B$ denote the event that the two M’s are next to each other.

Arrangements with Objects Separated

Counting with Inclusion-Exclusion

Permutations with Identical Objects

Arrangements with Objects Together

Complementary Counting

2021 RI P2 Q8

Permutation and Combination

<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 $\mathrm{P}\left( B\cap C \right)=0.14$.

2023 JPJC P2 Q8

<p>$A$ and $B$ are independent events such that $\mathrm{P}\left( \left. A \right|B \right)=\frac{1}{5}$and $\mathrm{P}\left( A\cup B \right)=\frac{2}{3}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the exact value of $\mathrm{P}\left( A\cap B' \right)$.</div><p style="text-align: right;">[4]</p></li></ol>
<p>For a third event $C$ , it is given that $\mathrm{P}\left( C \right)=\frac{2}{7}$ and that $A$ and $C$ are mutually exclusive.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the range of values of $\mathrm{P}\left( B'\cap C' \right)$.</div><p style="text-align: right;">[3]</p></li></ol>

$A$ and $B$ are independent events such that $\mathrm{P}\left( \left. A \right|B \right)=\frac{1}{5}$and $\mathrm{P}\left( A\cup B \right)=\frac{2}{3}$.

For a third event $C$ , it is given that $\mathrm{P}\left( C \right)=\frac{2}{7}$ and that $A$ and $C$ are mutually exclusive.

2021 TMJC P2 Q7

<p>For events $A$,$B$ and $C$ it is given that $\mathrm{P}\left( A \right)=0.8$, $\mathrm{P}\left( B \right)=0.4$, $\mathrm{P}\left( C \right)=c$ and $\mathrm{P}\left( A\cup B \right)=0.95$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find $\mathrm{P}\left( A\cap B \right)$ and determine with reason, if events $A$ and $B$ are independent.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">State the range of values of $c$ which necessarily implies that events $A$ and $C$ are not mutually exclusive.</div><p style="text-align: right;">[1]</p></li></ol>
<p>It is now given that $\mathrm{P}\left( C \right)=\mathrm{P}\left( \left. C \right|A \right)=0.45$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find $\mathrm{P}\left( A\cap C \right)$ and state the greatest and least possible values of $\mathrm{P}\left( A'\cap B\cap C \right)$.</div><p style="text-align: right;">[3]</p></li></ol>

For events $A$,$B$ and $C$ it is given that $\mathrm{P}\left( A \right)=0.8$, $\mathrm{P}\left( B \right)=0.4$, $\mathrm{P}\left( C \right)=c$ and $\mathrm{P}\left( A\cup B \right)=0.95$.

It is now given that $\mathrm{P}\left( C \right)=\mathrm{P}\left( \left. C \right|A \right)=0.45$.

2021 VJC P2 Q7

<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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