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<p>In a survey of $140$ students who had recently migrated with their families to Singapore, students were asked to indicate the birth places of both their parents. The results are displayed in the accompanying table.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 500px;"><colgroup><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"></colgroup><tbody><tr style="height: 30px;"><td colspan="2" rowspan="2" style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center"></td><td colspan="3" style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">Mother’s birth place</td></tr><tr style="height: 30px;"><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">UK</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">Europe</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">Asia</td></tr><tr style="height: 30px;"><td rowspan="3" style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center"><p>Father’s birth place</p></td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center"><p>UK</p></td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$45$</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$16$</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$4$</td></tr><tr style="height: 30px;"><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center"><p>Europe</p></td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$7$</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$12$</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$1$</td></tr><tr style="height: 30px;"><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center"><p>Asia</p></td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$8$</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$2$</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$45$</td></tr></tbody></table></div></div>
<ol class="i-list"><li data-list-item-id="item-1760783275343-g48xz2e43" data-list-item-path="item-1760783275343-g48xz2e43"><div class="list-item-paragraph">Find the probability that a randomly selected student from this group has both parents born in the UK.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760783275831-gsbc0aj4s" data-list-item-path="item-1760783275831-gsbc0aj4s"><div class="list-item-paragraph">Find the probability that a student randomly selected from those with mothers born in the UK, has a father born in Asia.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760783276355-paukr8n5b" data-list-item-path="item-1760783276355-paukr8n5b"><div class="list-item-paragraph">Find the probability that a student randomly chosen from this group have a parent born in either UK or Europe.<span class="inline-marks">[3]</span></div></li></ol>

In a survey of $140$ students who had recently migrated with their families to Singapore, students were asked to indicate the birth places of both their parents. The results are displayed in the accompanying table.

Conditional Probability from Tables

Addition Rule for Overlapping Events

2015 NYJC P2 Q8

Probability

<p>John has a collection of $230$ Rego bricks. Rego bricks come in $3$ different sizes and $4$ different colours. Bricks of the same size and colour are identical. The number of Rego bricks of each size and colour are shown in the table below</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760165071540-g9zhajz6d14.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAxNjUwNzE1NDAtZzl6aGFqejZkMTQucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTYzMSwiZXhwIjoxNzkyNzQ5OTMxfQ.2wL5ZjgITKPIpEj7IkG9W4jTytzTEkV1ppYBI9IDmQU" width="400" class="aligncenter" /></figure>
<p>John puts all the Rego bricks in a bag and picks one brick at random.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-paragraph">Find the probability that this brick is large and red.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the probability that this brick is either small or white.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the probability that this brick is either medium or large, given that it is not blue.</div><p style="text-align: right;">[3]</p></li></ol></li></ol>
<p>John now puts all the Rego bricks back in the bag and chooses two bricks at random without replacement.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find the probability that the two chosen bricks include exactly one small brick and exactly one red brick.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Among the $12$ possible size and colour combinations, it is given that the probability of John choosing two bricks of $2$ particular size and colour combinations that are different is $\frac{11}{1145}$. Write down the size and colour combinations of these two bricks.</div><p style="text-align: right;">[3]</p></li></ol>

John has a collection of $230$ Rego bricks. Rego bricks come in $3$ different sizes and $4$ different colours. Bricks of the same size and colour are identical. The number of Rego bricks of each size and colour are shown in the table below

John puts all the Rego bricks in a bag and picks one brick at random.

John now puts all the Rego bricks back in the bag and chooses two bricks at random without replacement.

Selections with Restrictions

Probability Using Permutations and Combinations

2023 HCI J2 MYE P2 Q10

Permutation and Combination

<ol class="abc-list"><li><div class="list-item-paragraph">A public opinion poll surveyed a sample of 1000 voters. The table below shows the number of males and females supporting Party A, Party B and Party C.</div></li></ol>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 360px;"><colgroup><col style="width: 90px;"><col style="width: 90px;"><col style="width: 90px;"><col style="width: 90px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p><strong>Party A</strong></p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p><strong>Party B</strong></p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p><strong>Party C</strong></p></th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p><strong>Male</strong></p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$200$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$130$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$70$</p></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p><strong>Female</strong></p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$250$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$300$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$50$</p></td></tr></tbody></table></div></div>
<p>One of the voters is chosen at random. Events $A$, $C$ and $M$ are defined as follows:<br>$A$: The voter chosen supports Party A. <br>$C$: The voter chosen supports Party C. <br>$M$: The voter chosen is a male. <br>Find</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\mathrm{P}\left( A|M \right)$,</div></li><li><div class="list-item-paragraph">$\mathrm{P}\left( M'\cap C' \right)$.</div></li></ol>
<p>Determine whether $A$ and $M$ are independent.</p>

One of the voters is chosen at random. Events $A$, $C$ and $M$ are defined as follows:
$A$: The voter chosen supports Party A. 
$C$: The voter chosen supports Party C. 
$M$: The voter chosen is a male. 
Find

Determine whether $A$ and $M$ are independent.

Independent Events

Conditional Probability

2010 HCI P2 Q10

<p>A blind box series named MathBots includes $4$ bots and $4$ number sets as accessories for the bots. Bots are named $\alpha $, $\beta $ and $\gamma $ with $\omega $ as the special character bot.</p>
<p>The number sets are $\mathbb{R}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{N}$. Each box of the series contains one of the bots and one of the number sets.</p>
<p>The data in the table below shows the probabilities of getting boxes with the specified contents.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 500px;"><colgroup><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"></colgroup><thead><tr><th style="text-align: left; --cell-justify: flex-start"></th><th style="text-align: center; --cell-justify: center">$\mathbb{R}$</th><th style="text-align: center; --cell-justify: center">$\mathbb{Z}$</th><th style="text-align: center; --cell-justify: center">$\mathbb{Q}$</th><th style="text-align: center; --cell-justify: center">$\mathbb{N}$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$\alpha$</td><td style="text-align: center; --cell-justify: center">$\frac{1}{8}$</td><td style="text-align: center; --cell-justify: center">$\frac{1}{8}$</td><td style="text-align: center; --cell-justify: center">$0$</td><td style="text-align: center; --cell-justify: center">$\frac{1}{20}$</td></tr><tr><td style="text-align: center; --cell-justify: center">$\beta$</td><td style="text-align: center; --cell-justify: center">$0$</td><td style="text-align: center; --cell-justify: center">$\frac{1}{8}$</td><td style="text-align: center; --cell-justify: center">$\frac{1}{8}$</td><td style="text-align: center; --cell-justify: center">$\frac{1}{20}$</td></tr><tr><td style="text-align: center; --cell-justify: center">$\gamma$</td><td style="text-align: center; --cell-justify: center">$\frac{1}{8}$</td><td style="text-align: center; --cell-justify: center">$0$</td><td style="text-align: center; --cell-justify: center">$\frac{1}{8}$</td><td style="text-align: center; --cell-justify: center">$\frac{1}{20}$</td></tr><tr><td style="text-align: center; --cell-justify: center">$\omega$</td><td style="text-align: center; --cell-justify: center">$0$</td><td style="text-align: center; --cell-justify: center">$0$</td><td style="text-align: center; --cell-justify: center">$0$</td><td style="text-align: center; --cell-justify: center">$\frac{1}{10}$</td></tr></tbody></table></div></div>
<p>(For example, there is a $\frac{1}{10}$ probability that a randomly chosen box contains $\omega $ and $\mathbb{N}$.)</p>
<p>A collector randomly chooses $4$ boxes for purchase.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the probability of the collector getting all the bots and number sets is $\frac{3}{320}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the probability that the collector gets all the bots given he manages to get all the number sets.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Determine if the event that he gets all the bots and the event that he gets all the number sets are independent.<span class="inline-marks">[2]</span></div></li></ol>

A blind box series named MathBots includes $4$ bots and $4$ number sets as accessories for the bots. Bots are named $\alpha $, $\beta $ and $\gamma $ with $\omega $ as the special character bot.

The number sets are $\mathbb{R}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{N}$. Each box of the series contains one of the bots and one of the number sets.

The data in the table below shows the probabilities of getting boxes with the specified contents.

(For example, there is a $\frac{1}{10}$ probability that a randomly chosen box contains $\omega $ and $\mathbb{N}$.)

A collector randomly chooses $4$ boxes for purchase.

2023 RVHS P2 Q8

<p>The table below shows the number of male and female students studying Chemistry, Physics and Biology at a private school.</p>
<div class="content-block content-block--table"><div class="table-container"><table><colgroup><col style="width: 110px;"><col style="width: 110px;"><col style="width: 110px;"><col style="width: 110px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Chemistry</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Physics</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Biology</p></th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Male</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$200$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$130$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$70$</p></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Female</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$250$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$300$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$50$</p></td></tr></tbody></table></div></div>
<p>One of the students is chosen at random. Events $C$, $B$ and $M$ are defined as follows:</p>
<p>$C$ : The student chosen is studying Chemistry.</p>
<p>$B$ : The student chosen is studying Biology.</p>
<p>$M$ : The student chosen is a male.</p>
<p>Find</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\mathrm{P}\left( C|M \right)$,<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">$\mathrm{P}\left( M\cup C \right)$,<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">$\mathrm{P}\left( M'\cap B' \right)$.<span class="inline-marks">[1]</span></div></li></ol>
<p>Determine whether $C$ and $M$ are independent.<span class="inline-marks">[2]</span></p>
<p>It is given that $20\%$ of Chemistry students, $30\%$ of Physics students and $5\%$ of Biology students are international students.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">One of the students selected at random is an international student. What is the probability that this student studies Chemistry?<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Three students are chosen at random. Find the probability that there is exactly one international student who studies Physics.<span class="inline-marks">[2]</span></div></li></ol>

The table below shows the number of male and female students studying Chemistry, Physics and Biology at a private school.

One of the students is chosen at random. Events $C$, $B$ and $M$ are defined as follows:

$C$ : The student chosen is studying Chemistry.

$B$ : The student chosen is studying Biology.

Determine whether $C$ and $M$ are independent.

It is given that $20\%$ of Chemistry students, $30\%$ of Physics students and $5\%$ of Biology students are international students.

One of the students selected at random is an international student. What is the probability that this student studies Chemistry?

Three students are chosen at random. Find the probability that there is exactly one international student who studies Physics.

Math Question Bank