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>Ava and Ben take turns throwing a dart at a target. The probability that Ava hits the target on any throw is $\dfrac{2}{5}$, while the probability that Ben hits it on any throw is $\dfrac{1}{3}$. All throws are independent. A player wins as soon as that player hits the target. Ava throws first.</p>
<ol class="i-list"><li data-list-item-id="gp-q1-i" data-list-item-path="gp-q1-i"><div class="list-item-paragraph">Show that the probability that both players miss in one pair of consecutive throws, one by Ava followed by one by Ben, is $\dfrac{2}{5}$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="gp-q1-ii" data-list-item-path="gp-q1-ii"><div class="list-item-paragraph">Find the probability that Ava wins the game.<span class="inline-marks">[3]</span></div></li></ol>
<p>In a second game, a biased coin is used to choose who throws first. Ben is twice as likely as Ava to throw first. The same hit probabilities and rules apply.</p>
<ol class="i-list" start="3"><li data-list-item-id="gp-q1-iii" data-list-item-path="gp-q1-iii"><div class="list-item-paragraph">Find the probability that Ava wins the second game.<span class="inline-marks">[3]</span></div></li></ol>

Ava and Ben take turns throwing a dart at a target. The probability that Ava hits the target on any throw is $\dfrac{2}{5}$, while the probability that Ben hits it on any throw is $\dfrac{1}{3}$. All throws are independent. A player wins as soon as that player hits the target. Ava throws first.

In a second game, a biased coin is used to choose who throws first. Ben is twice as likely as Ava to throw first. The same hit probabilities and rules apply.

Repeated Trials with Infinitely Many Possible Outcomes

Probabilities of Successive Events

Law of Total Probability

Application of Sum to Infinity

Waiting-Time and Stopping-Rule Probabilities

Sequence and Series

Probability

<p>A company buys $p\%$ of its electronic components from supplier $A$ and the remaining $(100 - p)\%$ from supplier $B$. The probability that a randomly chosen component supplied by $A$ is faulty is $0.05$. The probability that a randomly chosen component supplied by $B$ is faulty is $0.03$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$p = 25$, find the probability that a randomly chosen component is faulty.</span></div></li><li><div class="list-item-paragraph">For a general value of $p$, the probability that a randomly chosen component that is faulty was supplied by $A$ is denoted by $\mathrm{f}(p)$. Show that&nbsp;$\mathrm{f}(p) = \frac{0.05p}{0.02p + 3}$. Prove by differentiation that&nbsp;$\mathrm{f}$ is an increasing function for&nbsp;$0 \leq p \leq 100$, and explain what this statement means in the context of the question.</div></li></ol>

A company buys $p\%$ of its electronic components from supplier $A$ and the remaining $(100 - p)\%$ from supplier $B$. The probability that a randomly chosen component supplied by $A$ is faulty is $0.05$. The probability that a randomly chosen component supplied by $B$ is faulty is $0.03$.

Given that $p = 25$, find the probability that a randomly chosen component is faulty.

Bayes' Theorem

Quotient Rule

Increasing and Decreasing Functions

N2009 P2 Q7

Techniques of Differentiation

Applications of Differentiation

<p>Mr Hsu takes the train to office for work every weekday and is supposed to arrive at office by $7.30$ a.m. On average, he reaches the train station at $6.00$ a.m. His arrival time at the train station is normally distributed with a standard deviation of $10$ minutes. Every morning, there are only two trains, each departing $5$ minutes apart, with the first train departing at $6.10$ a.m. sharp. The time taken for the train journey follows a normal distribution with mean $60$ minutes and standard deviation $4$ minutes.</p>
<p>After alighting from the train, the time taken to walk from the train station to his office follows a normal distribution with mean $10$ minutes and standard deviation $3$ minutes. Mr Hsu will be late for work if he misses the second train or arrives at office after $7.30$ a.m. Assume that all travelling and waiting times are independent of each other.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">On a randomly chosen day, find the probability that Mr Hsu takes the first train and is late for work.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">On a randomly chosen day, show that the probability that Mr Hsu is late for work is $0.101$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">On a randomly chosen day, Mr Hsu has a briefing to attend before work. Given that he takes the first train, find the earliest starting time of the briefing scheduled on a whole minute for which the probability that he is late for the briefing is not more than $0.1$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">In a randomly chosen year of $260$ work days, find the expected number of days that Mr Hsu arrive late for work.<span class="inline-marks">[2]</span></div></li></ol>

Mr Hsu takes the train to office for work every weekday and is supposed to arrive at office by $7.30$ a.m. On average, he reaches the train station at $6.00$ a.m. His arrival time at the train station is normally distributed with a standard deviation of $10$ minutes. Every morning, there are only two trains, each departing $5$ minutes apart, with the first train departing at $6.10$ a.m. sharp. The time taken for the train journey follows a normal distribution with mean $60$ minutes and standard deviation $4$ minutes.

After alighting from the train, the time taken to walk from the train station to his office follows a normal distribution with mean $10$ minutes and standard deviation $3$ minutes. Mr Hsu will be late for work if he misses the second train or arrives at office after $7.30$ a.m. Assume that all travelling and waiting times are independent of each other.

On a randomly chosen day, find the probability that Mr Hsu takes the first train and is late for work.

On a randomly chosen day, show that the probability that Mr Hsu is late for work is $0.101$.

On a randomly chosen day, Mr Hsu has a briefing to attend before work. Given that he takes the first train, find the earliest starting time of the briefing scheduled on a whole minute for which the probability that he is late for the briefing is not more than $0.1$.

In a randomly chosen year of $260$ work days, find the expected number of days that Mr Hsu arrive late for work.

Normal Probabilities and Quantiles

Linear Combinations of Independent Normal Variables

2024 DHS H1 Q11

Continuous Random Variables

<p>A game at a funfair involves pulling balls of different colours out of a bag. A bag contains $1$ white ball, $2$ blue balls, $3$ green balls and $4$ red balls. The aim of the game is to pull the white ball from the bag as many times as possible. Players get three attempts. If a red ball is drawn, it is discarded, but any other ball is replaced in the bag.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that a player draws a white ball and two green balls on their three attempts.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that exactly one white ball was drawn, given that a red ball was drawn on the second attempt.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>Events $A$ and $B$ are defined as</p>
<p style="margin-left: 14rem">$A$: A white ball is drawn on the first attempt<br>$B$: A white ball is drawn on the second attempt</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Determine with reason whether events $A$ and $B$ are independent.</span><span class="inline-marks">[2]</span></div></li></ol>

A game at a funfair involves pulling balls of different colours out of a bag. A bag contains $1$ white ball, $2$ blue balls, $3$ green balls and $4$ red balls. The aim of the game is to pull the white ball from the bag as many times as possible. Players get three attempts. If a red ball is drawn, it is discarded, but any other ball is replaced in the bag.

Find the probability that a player draws a white ball and two green balls on their three attempts.

Find the probability that exactly one white ball was drawn, given that a red ball was drawn on the second attempt.

$A$: A white ball is drawn on the first attempt
$B$: A white ball is drawn on the second attempt

Determine with reason whether events $A$ and $B$ are independent.

2025 RVHS P2 Q8

<p>A basketball free throw game involves a team of two players, Ben and John. The game consists of at most $3$ throws and the moment a player makes a successful shot, the team wins, and the game will end.</p>
<p>The game uses the following rules.</p>
<ul class="bullet-list" style="margin-left: 2.5rem"><li data-list-item-id="rule-1" data-list-item-path="rule-1"><div class="list-item-paragraph">Only one player is selected for each game and the probability that Ben is selected in a game is $0.7$.</div></li><li data-list-item-id="rule-2" data-list-item-path="rule-2"><div class="list-item-paragraph">The probabilities that Ben and John make a successful shot in any single attempt are $0.1$ and $0.07$ respectively.</div></li><li data-list-item-id="rule-3" data-list-item-path="rule-3"><div class="list-item-paragraph">The shots are independent of each other.</div></li></ul>
<ol class="abc-list"><li data-list-item-id="item-a-4" data-list-item-path="item-a-4"><div class="list-item-paragraph">Find the probability that the team wins the game.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b-4" data-list-item-path="item-b-4"><div class="list-item-paragraph">The team did not win the game. Find the probability that John was the one who was selected to shoot.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c-4" data-list-item-path="item-c-4"><div class="list-item-paragraph">The team attempts the game repeatedly until the first game is won. Find the least number of attempts required such that the probability of winning within $n$ games is at least $0.95$.<span class="inline-marks">[2]</span></div></li></ol>

A basketball free throw game involves a team of two players, Ben and John. The game consists of at most $3$ throws and the moment a player makes a successful shot, the team wins, and the game will end.

Only one player is selected for each game and the probability that Ben is selected in a game is $0.7$.

The probabilities that Ben and John make a successful shot in any single attempt are $0.1$ and $0.07$ respectively.

Find the probability that the team wins the game.

The team did not win the game. Find the probability that John was the one who was selected to shoot.

The team attempts the game repeatedly until the first game is won. Find the least number of attempts required such that the probability of winning within $n$ games is at least $0.95$.

2025 HCI P2 Q6

<p>Angel either walks or cycles to school every morning. The probability that she walks is $\frac{2}{3}$. When she walks, the probability that she is late for school is $\frac{4}{9}$. When she cycles, the probability that she is late for school is $\frac{1}{9}$. Find the probability that on a particular morning,</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Angel walks and is not late for school,</span></div></li><li><div class="list-item-content"><span class="list-item-text">she is late for school.</span></div></li></ol>

Angel either walks or cycles to school every morning. The probability that she walks is $\frac{2}{3}$. When she walks, the probability that she is late for school is $\frac{4}{9}$. When she cycles, the probability that she is late for school is $\frac{1}{9}$. Find the probability that on a particular morning,

Total Probability from Travel Choices

<p>In a class of $25$ students, $14$ are boys and $11$ are girls. A teacher selects a student at random from the class. Another student is selected at random from the remaining students.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="list-item-paragraph">Copy and complete the tree diagram below.</div></li><li><div class="list-item-paragraph">Find the probability that</div><ol class="i-list"><li><div class="list-item-paragraph">the first student selected is a girl, and the second student is a boy,</div></li><li><div class="list-item-paragraph">only one of the two students selected is a girl,</div></li><li><div class="list-item-paragraph">the second student selected is a boy.</div></li></ol></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1766144163628-ajcq1ng243l.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjYxNDQxNjM2MjgtYWpjcTFuZzI0M2wucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTc2OCwiZXhwIjoxNzkyNzUwMDY4fQ.TNGerAJDr5hto9co5EuXD5M1ZihnkMKjRc66yMIG3GA" style="max-width: min(250px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

In a class of $25$ students, $14$ are boys and $11$ are girls. A teacher selects a student at random from the class. Another student is selected at random from the remaining students.

Dependent Events

Constructing and Using Probability Trees

Probability Trees for Dependent Events

<p>There are $2$ bags, $A$ and $B$. $A$ contains $3$ red balls and $5$ green balls while $B$ contains $4$ red balls and $3$ green balls. A game is played by $2$ players where a fair $6$ sided die is first thrown.<br></p>
<p>If the die shows the numbers $1$ or $2$, the player picks a ball from Bag $A$ and places it in Bag $B$. He then picks a ball from Bag $B$. If the die shows $3$, $4$, $5$ or $6$, the player loses his turn. The other player will then throw the die. The player who picks a red ball from Bag $B$ wins the game. Mark and Anthony take turns to play the game with Mark starting first. After each person has played the game, the balls are placed back into their original bags and the game starts again.</p>
<p>The game stops when a player has won.</p>
<p>Find the probability that</p>
<ol class="abc-list"><li data-list-item-id="item-1766239416643-9a8qfpxv4" data-list-item-path="item-1766239416643-9a8qfpxv4"><div class="list-item-paragraph">Mark wins the game on the first try.</div></li><li data-list-item-id="item-1766239417625-iuw4tlw34" data-list-item-path="item-1766239417625-iuw4tlw34"><div class="list-item-paragraph">Anthony wins the game on his first try.</div></li></ol>
<p>Every ball selection is uniform at random. Each die throw is independent, and the bags are reset before the next turn if nobody has won.</p>

There are $2$ bags, $A$ and $B$. $A$ contains $3$ red balls and $5$ green balls while $B$ contains $4$ red balls and $3$ green balls. A game is played by $2$ players where a fair $6$ sided die is first thrown.


If the die shows the numbers $1$ or $2$, the player picks a ball from Bag $A$ and places it in Bag $B$. He then picks a ball from Bag $B$. If the die shows $3$, $4$, $5$ or $6$, the player loses his turn. The other player will then throw the die. The player who picks a red ball from Bag $B$ wins the game. Mark and Anthony take turns to play the game with Mark starting first. After each person has played the game, the balls are placed back into their original bags and the game starts again.

Every ball selection is uniform at random. Each die throw is independent, and the bags are reset before the next turn if nobody has won.

2020 ACSI EM S4 P1

<p>In a hypothetical population, $2\%$ have a disease. A test returns positive for&nbsp;$90\%$ of those with the disease and negative for&nbsp;$90\%$ of those without it. A person is selected uniformly at random from the population.</p>
<ol class="i-list"><li data-list-item-id="item-1766237404568-r3fvjtm8z" data-list-item-path="item-1766237404568-r3fvjtm8z"><div class="list-item-paragraph">Find the probability that a person tests positive.</div></li><li data-list-item-id="item-1766237405745-fzy69zbqi" data-list-item-path="item-1766237405745-fzy69zbqi"><div class="list-item-paragraph">It is given that person tests positive. What is the chance they actually have the disease?</div></li></ol>

In a hypothetical population, $2\%$ have a disease. A test returns positive for $90\%$ of those with the disease and negative for $90\%$ of those without it. A person is selected uniformly at random from the population.

<p>In a game of chance, a player has to spin a fair spinner. The spinner has $7$ sections and an arrow which has an equal chance of coming to rest over any of the $7$ sections. The spinner has $1$ section labelled $\mathbf{R}$, $2$ sections labelled $\mathbf{B}$ and $4$ sections labelled $\mathbf{Y}$ (see diagram).</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760955584642-1qpwfpsbvyoi.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA5NTU1ODQ2NDItMXFwd2Zwc2J2eW9pLnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTI3NDk3NjgsImV4cCI6MTc5Mjc1MDA2OH0.VmvfGP4bUKKAkoBk5OROma7jDIfYemRYJ6Dw6qxfPWE" width="400" class="aligncenter" /></figure>
<p>The player then has to throw one of three fair six-sided dice, coloured red, blue or yellow. If the spinner comes to rest over $\mathbf{R}$ the red die is thrown, if the spinner comes to rest over $\mathbf{B}$ the blue die is thrown and if the spinner comes to rest over $\mathbf{Y}$ the yellow die is thrown. The yellow die has one face with $*$ on it, the blue die has two faces with $*$ on it and the red die has three faces with $*$ on it. The player wins the game if the die thrown comes to rest with a face showing $*$ uppermost.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the probability that a player wins a game.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Given that a player wins a game, find the probability that the spinner came to rest over $B$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the probability that a player wins $3$ consecutive games, each time throwing a die of a different colour.</div><p style="text-align: right;">[2]</p></li></ol>

In a game of chance, a player has to spin a fair spinner. The spinner has $7$ sections and an arrow which has an equal chance of coming to rest over any of the $7$ sections. The spinner has $1$ section labelled $\mathbf{R}$, $2$ sections labelled $\mathbf{B}$ and $4$ sections labelled $\mathbf{Y}$ (see diagram).

The player then has to throw one of three fair six-sided dice, coloured red, blue or yellow. If the spinner comes to rest over $\mathbf{R}$ the red die is thrown, if the spinner comes to rest over $\mathbf{B}$ the blue die is thrown and if the spinner comes to rest over $\mathbf{Y}$ the yellow die is thrown. The yellow die has one face with $*$ on it, the blue die has two faces with $*$ on it and the red die has three faces with $*$ on it. The player wins the game if the die thrown comes to rest with a face showing $*$ uppermost.

Probability Using Permutations and Combinations

Conditional Probability from Tree Diagrams

N2016 P2 Q5

<p>A dart player, Mike, in Smart Andrew’s College Darts Team is having a practice session of dart throwing in the hall. The probability that Mike hits his target on the first throw is $0.7$.</p>
<p>For each throw after his first throw, the probability that Mike hits the target, given that he hit the target on his preceding throw, is $0.77$.</p>
<p>For each throw after his first throw, the probability that Mike does not hit the target, given that he did not hit the target on his preceding throw, is $0.33$.</p>
<ol class="i-list"><li data-list-item-id="item-1760783192258-ct6klzlos" data-list-item-path="item-1760783192258-ct6klzlos"><div class="list-item-paragraph">Draw a fully labelled tree diagram to represent Mike’s first two throws.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760783192993-31duwnd34" data-list-item-path="item-1760783192993-31duwnd34"><div class="list-item-paragraph">Find the probability that Mike hits his target on the second throw of the dart.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is now known that Mike threw the dart three times.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760783204528-f16jotcy9" data-list-item-path="item-1760783204528-f16jotcy9"><div class="list-item-paragraph">Given that Mike did not hit his target on the second throw, find the probability that he hit the target twice.<span class="inline-marks">[3]</span></div></li></ol>

A dart player, Mike, in Smart Andrew’s College Darts Team is having a practice session of dart throwing in the hall. The probability that Mike hits his target on the first throw is $0.7$.

For each throw after his first throw, the probability that Mike hits the target, given that he hit the target on his preceding throw, is $0.77$.

For each throw after his first throw, the probability that Mike does not hit the target, given that he did not hit the target on his preceding throw, is $0.33$.

It is now known that Mike threw the dart three times.

Conditional Probability

2024 SAJC P2 Q5

<p>$16\%$ of all phone calls are scam calls. A company designed a $2$-stage filtering anti-scam app, ${{X}^{2}}$scam, that can block scam calls which plague mobile phone users. In a pre-launch test, it is found that ${{X}^{2}}$scam blocked $88\%$ of calls that are scam calls and $5\%$ of calls that are non-scam calls in the first stage of filtering. Of the calls that are unblocked in the first stage, ${{X}^{2}}$scam blocks $99\%$of scam calls and $5\%$ of non-scam calls within the second stage. Calls that are blocked in the first stage do not go through a second stage of filtering.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Draw a tree diagram to represent the above information.</div><p style="text-align: right;">[2]</p></li></ol>
<p>A call is selected at random. Find the probability that</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">it is a scam call that is blocked.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">it is a scam call, given that the call is blocked.</div><p style="text-align: right;">[2]</p></li></ol>
<p>Three calls were selected at random.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find the probability that ${{X}^{2}}$scam correctly filters exactly two of the calls.</div><p style="text-align: right;">[4]</p></li></ol>

$16\%$ of all phone calls are scam calls. A company designed a $2$-stage filtering anti-scam app, ${{X}^{2}}$scam, that can block scam calls which plague mobile phone users. In a pre-launch test, it is found that ${{X}^{2}}$scam blocked $88\%$ of calls that are scam calls and $5\%$ of calls that are non-scam calls in the first stage of filtering. Of the calls that are unblocked in the first stage, ${{X}^{2}}$scam blocks $99\%$of scam calls and $5\%$ of non-scam calls within the second stage. Calls that are blocked in the first stage do not go through a second stage of filtering.

A call is selected at random. Find the probability that

Binomial Probabilities Using the Probability Mass Function

2022 EJC J2 MYE P2 Q9

Discrete Random Variables (DRV)

<p>It is known that in a certain country, $30\%$ of the people support Party A; $50\%$ of the people support Party B; and $20\%$ of the people support Party C.</p>
<p>In the recent election, $55\%$ of Party A supporters voted, as did $86\%$ of the supporters of Party B and $42\%$ of the supporters of Party C.</p>
<ol class="abc-list"><li data-list-item-id="item-1760434394931-tarcz50ty" data-list-item-path="item-1760434394931-tarcz50ty"><div class="list-item-paragraph">One person is selected at random. Find the probability that</div><ol class="i-list"><li data-list-item-id="subitem-1760434405739-kzhw65rfc" data-list-item-path="item-1760434394931-tarcz50ty/subitem-1760434405739-kzhw65rfc"><div class="list-item-paragraph">the person voted,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760434412008-t3wfs4g6h" data-list-item-path="item-1760434394931-tarcz50ty/subitem-1760434412008-t3wfs4g6h"><div class="list-item-paragraph">the person is a supporter of Party B, given that the person voted.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-1760434395741-gus7rlq94" data-list-item-path="item-1760434395741-gus7rlq94"><div class="list-item-paragraph">Two persons are selected at random. Find the probability that exactly one of them voted.<span class="inline-marks">[2]</span></div></li></ol>

It is known that in a certain country, $30\%$ of the people support Party A; $50\%$ of the people support Party B; and $20\%$ of the people support Party C.

In the recent election, $55\%$ of Party A supporters voted, as did $86\%$ of the supporters of Party B and $42\%$ of the supporters of Party C.

2023 VJC P2 Q6

<p>Diabetes is present in $27\%$ of the elderly population. In a health screening exercise, a quick test for diabetes is used, but the test is not totally reliable. It has&nbsp;$0.98$ probability of giving a positive result when an elderly has diabetes, and $0.04$ probability of giving a positive result when an elderly does not have diabetes. A randomly chosen elderly is given the test. Find the probability that</p>
<ol class="i-list"><li data-list-item-id="item-1760431941810-k5ultf6kk" data-list-item-path="item-1760431941810-k5ultf6kk"><div class="list-item-paragraph">the result of the test is positive,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760431942403-xdfkx3fbm" data-list-item-path="item-1760431942403-xdfkx3fbm"><div class="list-item-paragraph">the elderly has diabetes given that the result of the test is positive.<span class="inline-marks">[2]</span></div></li></ol>

Diabetes is present in $27\%$ of the elderly population. In a health screening exercise, a quick test for diabetes is used, but the test is not totally reliable. It has $0.98$ probability of giving a positive result when an elderly has diabetes, and $0.04$ probability of giving a positive result when an elderly does not have diabetes. A randomly chosen elderly is given the test. Find the probability that

2013 IJC P2 Q5

<p>A certain disease is present in $1$ in $200$ of the population. In a mass screening programme, a quick test for the disease is used but the test is not totally reliable. For someone who does have the disease there is a probability of $0.9$ that the test will prove positive, whereas for someone who does not have the disease, there is a probability of $0.02$ that the test will prove positive.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">A person is chosen at random and tested.</div><div class="list-item-image"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760417060035-89gldovncl5.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA0MTcwNjAwMzUtODlnbGRvdm5jbDUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTc2OCwiZXhwIjoxNzkyNzUwMDY4fQ.F9j4kiq5_j-y-rqvEB5wiQC0KKUiCL9tzUuHaB4qZno" alt="List item image" /></div><ol class="i-list"><li><div class="list-item-paragraph">Copy and complete the tree diagram which illustrates the application of one test.</div></li><li><div class="list-item-paragraph">Find the probability that the person has the disease and the test is positive.</div></li><li><div class="list-item-paragraph">Find the probability that the test is negative.</div></li><li><div class="list-item-paragraph">Given that the test is positive, find the probability that the person has the disease.</div></li></ol></li><li><div class="list-item-paragraph">People for whom the test proves positive are recalled and re-tested. Find the probability that a person has the disease if the second test also proves positive.</div></li></ol>

A certain disease is present in $1$ in $200$ of the population. In a mass screening programme, a quick test for the disease is used but the test is not totally reliable. For someone who does have the disease there is a probability of $0.9$ that the test will prove positive, whereas for someone who does not have the disease, there is a probability of $0.02$ that the test will prove positive.

N1997 P2 Q3

<p>Urn $1$ contains $5$ black balls and $3$ red balls. Urn $2$ contains $4$ black balls and $4$ red balls. An experiment is conducted in the following manner:</p>
<p>A fair die is tossed. If it shows $1$ or $2$, two balls are drawn from urn $1$ without replacement.</p>
<p>Otherwise, one ball is drawn from each urn. Let $X$ be the number of red balls drawn in one experiment.</p>
<p>Show that $\mathrm{P}\left( X=0 \right)=\frac{55}{168}$.</p><p style="text-align: right;">[1]</p>
<p>Find the probability distribution of $X$. Evaluate the expectation and variance of $X$.</p><p style="text-align: right;">[6]</p>

Urn $1$ contains $5$ black balls and $3$ red balls. Urn $2$ contains $4$ black balls and $4$ red balls. An experiment is conducted in the following manner:

A fair die is tossed. If it shows $1$ or $2$, two balls are drawn from urn $1$ without replacement.

Otherwise, one ball is drawn from each urn. Let $X$ be the number of red balls drawn in one experiment.

Show that $\mathrm{P}\left( X=0 \right)=\frac{55}{168}$.

Find the probability distribution of $X$. Evaluate the expectation and variance of $X$.

Constructing and Using Discrete Probability Distributions

Expectation and Variance of Discrete Random Variables

2005 TJC P2

<p>During an online mathematics test, multiple-choice questions are randomly selected from a large item bank. The probability that a selected question is on algebra is $\frac{4}{5}$ and that it is on geometry is $\frac{1}{5}$. Each question has four options, among which only one is correct. A candidate Amy, who has studied algebra but not geometry, answers each algebra question correctly with a probability of $\frac{7}{8}$. When faced with a geometry question, she simply makes a random guess. Four marks are awarded for each question answered correctly and one mark deducted for each question answered wrongly.</p>
<ol class="abc-list"><li data-list-item-id="item-1760342169030-mwds9xis5" data-list-item-path="item-1760342169030-mwds9xis5"><div class="list-item-paragraph">Consider a question attempted by Amy. Show that the probability that she answered the question correctly is $\frac{3}{4}$.</div></li><li data-list-item-id="item-1760342169720-21aoahzze" data-list-item-path="item-1760342169720-21aoahzze"><div class="list-item-paragraph">The random variable $X$ is the score obtained by Amy for a question. Write down the probability distribution of $X$.</div></li><li data-list-item-id="item-1760342170275-zvk5z4bjo" data-list-item-path="item-1760342170275-zvk5z4bjo"><div class="list-item-paragraph">Assume that Amy answers $50$ questions independently.</div><div class="list-item-paragraph">If Amy obtained at least $130$ marks, find the least number of questions that she answers correctly. Hence find the probability that Amy will obtain at least $130$ marks.</div></li></ol>

During an online mathematics test, multiple-choice questions are randomly selected from a large item bank. The probability that a selected question is on algebra is $\frac{4}{5}$ and that it is on geometry is $\frac{1}{5}$. Each question has four options, among which only one is correct. A candidate Amy, who has studied algebra but not geometry, answers each algebra question correctly with a probability of $\frac{7}{8}$. When faced with a geometry question, she simply makes a random guess. Four marks are awarded for each question answered correctly and one mark deducted for each question answered wrongly.

Binomial Probabilities Using Technology

Linear Combinations of Random Variables

2006 AJC P2 [Modified]

<p>Andrew plays a game with 4 cards numbered 1, 2 , 3 and 4. In each round, a random sample of $n$ cards is taken without replacement, where $1\le n\le 4$ and there is equal chance of Andrew taking 1, 2, 3 or 4 cards. Let $L$ denote the smallest value of the chosen cards. The distribution table for $L$ is as follows:</p>
<table><thead><tr><th>$L$</th><th>$1$</th><th>$2$</th><th>$3$</th><th>$4$</th></tr></thead><tbody><tr><td>${\mathbb{P}(L = l)}$</td><td>$a$</td><td>$b$</td><td>${\frac{5}{48}}$</td><td>${\frac{1}{16}}$</td></tr></tbody></table>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $b=\frac{5}{24}$.</div></li><li><div class="list-item-paragraph">Show that $\mathrm{Var}\left( L \right)=\frac{1895}{2304}$.</div></li><li><div class="list-item-paragraph">Andrew invites Jasmine to join in the game. In each round, only 1 player plays and the cards are replaced before the other player plays the game. Jasmine has the first move in the game. The first player to obtain the results $L=1$ will win the game. Find the probability that Jasmine wins the game.</div></li></ol>

Andrew plays a game with 4 cards numbered 1, 2 , 3 and 4. In each round, a random sample of $n$ cards is taken without replacement, where $1\le n\le 4$ and there is equal chance of Andrew taking 1, 2, 3 or 4 cards. Let $L$ denote the smallest value of the chosen cards. The distribution table for $L$ is as follows:

Determining Unknowns in Discrete Probability Distributions

2006 HCI P2 Q28 [Modified]

<p>In this question, you should state the parameters of any distributions that you use.</p>
<p>A stationery factory manufactures pens for sale. The diameters (in mm) of the pens have distribution $\mathrm{N}\left( \mathrm{10,0}\mathrm{.003} \right)$. The pens are packed in sets of $24$ into a box. Within the box, the pens are laid out side by side. The widths of the boxes (in mm) are normally distributed with mean $\mathrm{240}\mathrm{.5}$mm and variance $\mathrm{0}\mathrm{.02}$mm$^{2}$.</p>
<ol class="i-list"><li data-list-item-id="item-1760153813314-94hy215sv" data-list-item-path="item-1760153813314-94hy215sv"><div class="list-item-paragraph">Find the probability that a random sample of $24$ pens would fit into a randomly selected box.<span class="inline-marks">[3]</span></div></li></ol>
<p>A pen is considered “defective” if its diameter is not within $0.1$ mm of the population mean.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1760153848627-cqvfy7dk5" data-list-item-path="item-1760153848627-cqvfy7dk5"><div class="list-item-paragraph">Find the probability that a randomly selected pen is defective.<span class="inline-marks">[2]</span></div></li></ol>
<p>Pens are produced and inspected in batches. For each batch, a sample of $12$ pens is randomly selected and checked.</p>
<p>• If there are fewer than $2$ defective pens in this sample of $12$, the batch passes the inspection.</p>
<p>• If there are exactly $2$ defective pens in this sample, a second sample of $12$ pens is randomly selected and checked. If there are no defective pens in the second sample, the batch passes the inspection.</p>
<p>• Otherwise, the batch does not pass the inspection.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760153899663-oy4e56vyc" data-list-item-path="item-1760153899663-oy4e56vyc"><div class="list-item-paragraph">Show that the probability of a batch of pens passing the inspection is $\mathrm{0}\mathrm{.871}$, correct to $3$ significant figures.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760153900704-rtdze6eow" data-list-item-path="item-1760153900704-rtdze6eow"><div class="list-item-paragraph">Find the probability that not more than $3$ defective pens were found in a batch of pens during the inspection process, given that the batch of pens did not pass the inspection.<span class="inline-marks">[3]</span></div></li></ol>
<p>The factory also manufactures and packs pencils and crayons in boxes of $12$ each. During a promotion period, special pencils and special crayons are produced and randomly included in some boxes. The number of special pencils in a box of pencils has distribution $\mathrm{B}\left( 12,0.07 \right)$, while the number of special crayons in a box of crayons has distribution $\mathrm{B}\left( \mathrm{12, 0}\mathrm{.06} \right)$.</p>
<ol class="i-list" start="5"><li data-list-item-id="item-1760153950295-7mpqt7a2w" data-list-item-path="item-1760153950295-7mpqt7a2w"><div class="list-item-paragraph">$40$ boxes of pencils and $40$ boxes of crayons are randomly selected. Find the probability that the mean number of special pencils per box of pencils is more than the mean number of special crayons per box of crayons.<span class="inline-marks">[4]</span></div></li></ol>

In this question, you should state the parameters of any distributions that you use.

A stationery factory manufactures pens for sale. The diameters (in mm) of the pens have distribution $\mathrm{N}\left( \mathrm{10,0}\mathrm{.003} \right)$. The pens are packed in sets of $24$ into a box. Within the box, the pens are laid out side by side. The widths of the boxes (in mm) are normally distributed with mean $\mathrm{240}\mathrm{.5}$mm and variance $\mathrm{0}\mathrm{.02}$mm$^{2}$.

A pen is considered “defective” if its diameter is not within $0.1$ mm of the population mean.

Pens are produced and inspected in batches. For each batch, a sample of $12$ pens is randomly selected and checked.

• If there are fewer than $2$ defective pens in this sample of $12$, the batch passes the inspection.

• If there are exactly $2$ defective pens in this sample, a second sample of $12$ pens is randomly selected and checked. If there are no defective pens in the second sample, the batch passes the inspection.

• Otherwise, the batch does not pass the inspection.

The factory also manufactures and packs pencils and crayons in boxes of $12$ each. During a promotion period, special pencils and special crayons are produced and randomly included in some boxes. The number of special pencils in a box of pencils has distribution $\mathrm{B}\left( 12,0.07 \right)$, while the number of special crayons in a box of crayons has distribution $\mathrm{B}\left( \mathrm{12, 0}\mathrm{.06} \right)$.

Central Limit Theorem

Normal Models for Sample Means and Averages

Comparing Independent Sample Means

2021 EJC P2 Q10

Sampling

<p>Every morning, Peter has to take a bus to school.On average, he reaches the bust stop at 6:30am each day.His arrival time at the bus stop is normally distributed with a standard deviation of 10 minutes.The first bus will leave the bust stop at 6:40am sharp and the second bus will leave at 6:50am sharp.Regardless of the time the busses leave the bus stop, the bus journey will follow a normal distribution with mean 45 minutes and standard deviation 20 minutes.Peter will be late if he arrives at school after 7:30 am or if he misses the 6:50 am bus.</p>
<p>Assuming that the time Peter reaches the bus stop and the time taken for the bus journey are independent, find the probability that</p>
<ol class="i-list"><li><div class="list-item-paragraph">Peter will miss the 6:50 am bus.</div></li><li><div class="list-item-paragraph">Show the probability that Peter will be late for school is 0.442.</div></li></ol>

Every morning, Peter has to take a bus to school.On average, he reaches the bust stop at 6:30am each day.His arrival time at the bus stop is normally distributed with a standard deviation of 10 minutes.The first bus will leave the bust stop at 6:40am sharp and the second bus will leave at 6:50am sharp.Regardless of the time the busses leave the bus stop, the bus journey will follow a normal distribution with mean 45 minutes and standard deviation 20 minutes.Peter will be late if he arrives at school after 7:30 am or if he misses the 6:50 am bus.

Assuming that the time Peter reaches the bus stop and the time taken for the bus journey are independent, find the probability that

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