A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>For events $A$ and $B$, it is given that&nbsp;$\mathrm{P}(A) = 0.6$, $\mathrm{P}(A \cup B) = 0.7$ and $\mathrm{P}(B \mid A) = 0.3$.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-1-item-0" data-list-item-path="imported-item-block-1-item-0"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{P}(B) = 0.28$ and determine with a reason, if&nbsp;$A$ and $B$ are independent.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-1-item-1" data-list-item-path="imported-item-block-1-item-1"><div class="list-item-paragraph">An event $C$ is such that&nbsp;$A$ and $C$ are mutually exclusive and $\mathrm{P}(C) &gt; 0.25$. Find the range of values of $\mathrm{P}(B' \cap C)$.<span class="inline-marks">[3]</span></div></li></ol>

For events $A$ and $B$, it is given that $\mathrm{P}(A) = 0.6$, $\mathrm{P}(A \cup B) = 0.7$ and $\mathrm{P}(B \mid A) = 0.3$.

Show that $\mathrm{P}(B) = 0.28$ and determine with a reason, if $A$ and $B$ are independent.

An event $C$ is such that $A$ and $C$ are mutually exclusive and $\mathrm{P}(C) > 0.25$. Find the range of values of $\mathrm{P}(B' \cap C)$.

Probability Bounds and Inequalities

Addition Rule for Overlapping Events

Probability

<p>The events $A$ and $B$ satisfy $P(A)=\dfrac{2}{5}$, $P(B')=\dfrac{3}{4}$ and $P(B\mid A)=\dfrac{1}{4}$.</p>
<ol class="i-list"><li data-list-item-id="prob-b-item-i" data-list-item-path="prob-b-item-i"><div class="list-item-paragraph">Find $P(A'\cap B')$ and hence prove that the events $A'$ and $B'$ are independent.<span class="inline-marks">[4]</span></div></li></ol>
<p>A third event $C$ satisfies $P(C)=\dfrac{1}{2}$ and $P(A\cap B\cap C)=\dfrac{1}{20}$. It is also known that&nbsp;$B$ and $C$ are independent, while $A$ and $C$ are not independent.</p>
<ol class="i-list" start="2"><li data-list-item-id="prob-b-item-ii" data-list-item-path="prob-b-item-ii"><div class="list-item-paragraph">Find the range of possible values of $P(A'\cap B'\cap C')$.<span class="inline-marks">[5]</span></div></li></ol>

The events $A$ and $B$ satisfy $P(A)=\dfrac{2}{5}$, $P(B')=\dfrac{3}{4}$ and $P(B\mid A)=\dfrac{1}{4}$.

A third event $C$ satisfies $P(C)=\dfrac{1}{2}$ and $P(A\cap B\cap C)=\dfrac{1}{20}$. It is also known that $B$ and $C$ are independent, while $A$ and $C$ are not independent.

Independent Events

Complementary Events

Conditional Probability

Applying Probability Laws

<p>For events $A$ and $B$, it is given that&nbsp;$P(A)=\dfrac{1}{4}$, $P(A\cup B)=\dfrac{5}{8}$ and $P(A\cap B')=\dfrac{1}{8}$.</p>
<ol class="i-list"><li data-list-item-id="prob-a-item-i" data-list-item-path="prob-a-item-i"><div class="list-item-paragraph">Find $P(B)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="prob-a-item-ii" data-list-item-path="prob-a-item-ii"><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;$P(C)=\dfrac{2}{5}$, $P(A\cap B\cap C)=\dfrac{1}{20}$, and $C$ is independent of $A$.</p>
<ol class="i-list" start="3"><li data-list-item-id="prob-a-item-iii" data-list-item-path="prob-a-item-iii"><div class="list-item-paragraph">Find $P(A'\cap C)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="prob-a-item-iv" data-list-item-path="prob-a-item-iv"><div class="list-item-paragraph">Find the range of possible values of $P(A'\cap B'\cap C')$.<span class="inline-marks">[3]</span></div></li></ol>

For events $A$ and $B$, it is given that $P(A)=\dfrac{1}{4}$, $P(A\cup B)=\dfrac{5}{8}$ and $P(A\cap B')=\dfrac{1}{8}$.

For a third event $C$, it is given that $P(C)=\dfrac{2}{5}$, $P(A\cap B\cap C)=\dfrac{1}{20}$, and $C$ is independent of $A$.

<p>Events $A$ and $B$ are such that&nbsp;$\mathrm{P}(A)=0.74$ and $\mathrm{P}(B)=0.45$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the range of values of $\mathrm{P}(A \cap B)$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the range of values of $\mathrm{P}(A \cup B)$.</span></div></li></ol>

Events $A$ and $B$ are such that $\mathrm{P}(A)=0.74$ and $\mathrm{P}(B)=0.45$.

Find the range of values of $\mathrm{P}(A \cap B)$.

Find the range of values of $\mathrm{P}(A \cup B)$.

IP2U10 HOT Q3

<p>The events $A$, $B$ and $C$ are such that $\mathrm{P}(A)=0.8$, $\mathrm{P}(B)=0.2$ and $\mathrm{P}(C)=0.6$. It is also known that $\mathrm{P}(A \cap B)=\mathrm{P}(B)$ and $\mathrm{P}(B \cap C)=0.1$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find exactly the maximum and minimum possible values of</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(A \cap B' \cap C)$,</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(A \cap B' \cap C | A \cup C)$.</span><span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">It is given further that $A$ and $C$ are independent, find the value of $\mathrm{P}(A \cap B' \cap C)$.</span><span class="inline-marks">[2]</span></div></li></ol>

The events $A$, $B$ and $C$ are such that $\mathrm{P}(A)=0.8$, $\mathrm{P}(B)=0.2$ and $\mathrm{P}(C)=0.6$. It is also known that $\mathrm{P}(A \cap B)=\mathrm{P}(B)$ and $\mathrm{P}(B \cap C)=0.1$.

Find exactly the maximum and minimum possible values of

It is given further that $A$ and $C$ are independent, find the value of $\mathrm{P}(A \cap B' \cap C)$.

2025 DHS P2 Q6

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg2NDExMzYzNzItNGNhYnV4dHh2encucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTY3NiwiZXhwIjoxNzkyNzQ5OTc2fQ.XUE6y1m5_FifMzYvbmsfnfatw1bxX0RlsOOMY81rsFs" 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 from Venn Diagrams

2025 CJC P2 Q6

<ol class="abc-list"><li><div class="list-item-paragraph">${A}$, ${B}$ and ${C}$ are taking part in a game in which a prize is hidden in one of $n$ identical closed boxes, where&nbsp;$n &gt; 4$. First, ${A}$ opens at random two boxes in an attempt to find the prize. If&nbsp;${A}$ fails to find the prize, ${B}$ then opens at random two of the remaining boxes in an attempt to find the prize. If&nbsp;${B}$ fails to find the prize, ${C}$ opens all the remaining boxes and finds the prize.</div><ol class="i-list"><li><div class="list-item-paragraph">Find, in terms of $n$, the probability that&nbsp;${A}$ finds the prize.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Show that the probability that B finds the prize is $\frac{2}{n}$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Find the range of values of $n$ for which the probability that&nbsp;${C}$ finds the prize is more than $8$ times the probability that either ${A}$ or ${B}$ find the prize.<span class="inline-marks">[3]</span></div></li></ol></li><li><div class="list-item-paragraph">The events $Q$ and $R$ are such that&nbsp;${P}\left( Q\cap R \right)=x$, $\mathrm{P}\left( Q\cap {R}' \right)=0.1$, ${P}\left( {Q}'\cap R \right)=\mathrm{P}\left( Q \right)$ and ${P}\left( \left( Q\cup R \right)' \right)=\mathrm{P}\left( Q \right)+2\mathrm{P}\left( R \right)$. By using a Venn diagram, or otherwise, find the exact value of $x$.<span class="inline-marks">[3]</span></div></li></ol>

Probabilities of Successive Events

Finding Unknowns in Probability Models

N2024 P2 Q7

<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 $A$.

Mutually Exclusive 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

<ol class="abc-list"><li><div class="list-item-paragraph">In a certain university, there are $110$ Co-Curricular Activities (CCAs) clustered into $4$ categories. There are $22$ Arts and Culture CCAs, $16$ Community Service CCAs, $38$ Physical Sports CCAs and $34$ Special Interest CCAs.</div><ol class="i-list"><li><div class="list-item-paragraph">Albert wishes to find out about approaches to training of the Physical Sports CCAs, so he sends a questionnaire to $22$ Physical Sports CCAs. Explain whether these $22$ Physical Sports CCAs form a sample or a population.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Benedict wishes to investigate the level of student engagement in CCAs, but does not want to obtain the detailed information necessary from all $110$ CCAs. Explain how he should carry out his investigation, and why he should do the investigation in this way.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the number of different possible samples of $16$ CCAs, with $4$ CCAs chosen from each category.<span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-paragraph">The probability of events $A$, $B$ and $C$ occurring are given by $\mathrm{P}\left( A \right)=\frac{8}{25}$, $\mathrm{P}\left( B \right)=\frac{61}{100}$ and $\mathrm{P}\left( C \right)=\frac{7}{20}$ respectively. It is also known that&nbsp;$\mathrm{P}\left( A\cap C \right)=\frac{11}{100}$, $\mathrm{P}\left( \left. B \right|C \right)=\frac{13}{35}$ and $\mathrm{P}\left( A\cap B\cap C \right)=\frac{7}{100}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Find $\mathrm{P}\left( B\cap C \right)$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">If events $A$ and $B$ are independent, find $\mathrm{P}\left( A\cap B\cap C' \right)$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">If events $A$ and $B$ are not known to be independent, find the greatest and least possible values of $\mathrm{P}\left( A\cap B\cap C' \right)$.<span class="inline-marks">[4]</span></div></li></ol></li></ol>

Sampling Techniques and Bias

Selections with Restrictions

Population and Random Samples

2024 CJC P2 Q10

Sampling

Permutation and Combination

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

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>The events $A$ and $B$ are such that $\mathrm{P}\left( A \right)=0.3$, $\mathrm{P}\left( B' \right)=0.85$ and $\mathrm{P}\left( \left. B \right|A \right)=0.15$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find $\mathrm{P}\left( A'\cap B' \right)$ and hence prove that the events $A'$ and $B'$ are independent events.</div><p style="text-align: right;">[4]</p></li></ol>
<p>A third event $C$ is such that $\mathrm{P}\left( C \right)=0.45$ and $\mathrm{P}\left( A\cap B\cap C \right)=0.03$. It is also given that events $B$ and $C$ are independent, and that events $A$ and $C$ are not independent.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the range of possible values of $\mathrm{P}\left( A'\cap B'\cap C' \right)$.</div><p style="text-align: right;">[5]</p></li></ol>

The events $A$ and $B$ are such that $\mathrm{P}\left( A \right)=0.3$, $\mathrm{P}\left( B' \right)=0.85$ and $\mathrm{P}\left( \left. B \right|A \right)=0.15$.

A third event $C$ is such that $\mathrm{P}\left( C \right)=0.45$ and $\mathrm{P}\left( A\cap B\cap C \right)=0.03$. It is also given that events $B$ and $C$ are independent, and that events $A$ and $C$ are not independent.

2022 SAJC J2 CT Q9

<p>Players $A$ and $B$ compete in a racquet match consisting of at most $3$ sets. Each set is won by either player $A$ or $B$, and the match is won by the first person to win two sets. Player $A$ has a probability of $\frac{2}{3}$ of winning each of the first two sets. If the match goes into the third set, player $A$ has a probability of $p$ of winning this set.</p>
<ol class="i-list"><li><div class="list-item-paragraph">With the aid of a tree diagram, find in terms of $p$, the probability that player $A$ will win the match.</div></li><li><div class="list-item-paragraph">Deduce the range of values of the probability that player $A$ wins the match.</div></li><li><div class="list-item-paragraph">Show that the value of $p$ in order for the match to be fair is $\frac{1}{8}$.</div></li></ol>
<p>Given that player $A$ wins the match, find, in terms of $p$, the probability that he won the ${{2}^{\mathrm{nd}}}$ set.</p>

Players $A$ and $B$ compete in a racquet match consisting of at most $3$ sets. Each set is won by either player $A$ or $B$, and the match is won by the first person to win two sets. Player $A$ has a probability of $\frac{2}{3}$ of winning each of the first two sets. If the match goes into the third set, player $A$ has a probability of $p$ of winning this set.

Given that player $A$ wins the match, find, in terms of $p$, the probability that he won the ${{2}^{\mathrm{nd}}}$ set.

Constructing and Using Probability Trees

Conditional Probability from Tree Diagrams

Waiting-Time and Stopping-Rule Probabilities

2011 NYJC

<p>A room contains $n$ randomly chosen people</p>
<ol class="i-list"><li><div class="list-item-paragraph">Assume that a randomly chosen person is equally likely to have been born on any day of the week. The probability that the people in the room were all born on different days of the week is denoted by $P$.</div><ol class="abc-list"><li><div class="list-item-paragraph">Find $P$ in the case $n=3$.</div></li><li><div class="list-item-paragraph">Show that $P=\frac{120}{343}$ in the case $n=4$.</div></li></ol></li><li><div class="list-item-paragraph">Assume now that a randomly chosen person is equally likely to have been born in any month of the year. Find the smallest value of $n$ such that the probability that the people in the room were all born in different months of the year is less than $0.5$.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">Assume now that a randomly chosen person is equally likely to be born on any of the $365$ days in the year. It is given that, for the case $n=21$, the probability that the people were all born on different days of the year is $0.55631$, correct to $5$ places of decimals. Without the use of a graphic calculator, find the smallest value of $n$ such that at least two of the people were born on the same day of the year exceeds $\frac{1}{2}$.</div></li></ol>

A room contains $n$ randomly chosen people

Math Question Bank