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PSLE

PSLE Foundation Mathematics

<p>A private-hire vehicle firm, Euber, employs drivers to operate a fleet of cars. The mileage covered by an Euber driver on a weekday is denoted by $M$ km. A random sample of $60$ Euber drivers is taken and their mileage can be summarised by</p>
<p style="text-align: center">\[\sum(m-250) = 540 \text{ and}\hspace{0.5em} \sum(m-259)^2 = 8176.\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Calculate unbiased estimates of the population mean and variance.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>Assume that the population mean and variance follow the answers calculated in part <strong>(i)</strong>.</p>
<p>Random samples of $n$ Euber drivers are taken, where&nbsp;$n$ is large.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Over a long period of time, it is found that&nbsp;$25\%$ of the samples have a mean mileage of at least $260$ km. Find the value of $n$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain whether it is necessary to assume that&nbsp;$M$ follows a normal distribution in order for your calculations in part <strong>(ii)</strong> to hold.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">It is now given that&nbsp;$M$ is normally distributed with population variance $5^2$. Estimate the sample size needed so that Euber can be $95\%$ certain that the sample mean mileage will differ from the true mean by less than $500$m.</span><span class="inline-marks">[2]</span></div></li></ol>

A private-hire vehicle firm, Euber, employs drivers to operate a fleet of cars. The mileage covered by an Euber driver on a weekday is denoted by $M$ km. A random sample of $60$ Euber drivers is taken and their mileage can be summarised by

\[\sum(m-250) = 540 \text{ and } \sum(m-259)^2 = 8176.\]

Calculate unbiased estimates of the population mean and variance.

Assume that the population mean and variance follow the answers calculated in part **(i)**.

Random samples of $n$ Euber drivers are taken, where $n$ is large.

Over a long period of time, it is found that $25\%$ of the samples have a mean mileage of at least $260$ km. Find the value of $n$.

Explain whether it is necessary to assume that $M$ follows a normal distribution in order for your calculations in part **(ii)** to hold.

It is now given that $M$ is normally distributed with population variance $5^2$. Estimate the sample size needed so that Euber can be $95\%$ certain that the sample mean mileage will differ from the true mean by less than $500$m.

Estimating Parameters from Summarised and Coded Data

Central Limit Theorem

Solving Normal Probability Inequalities

Determining Sample Size from Probability Conditions

Normal Models for Sample Means and Averages

Sample Mean and Its Sampling Distribution

2019 EJC BT2 P2 Q8 [Modified]

Sampling

Continuous Random Variables

<p>When a large number of mangoes are harvested, $37$% of them are unripe. The mangoes are packed randomly in boxes of $6$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the probability that there is no unripe mango in a box is $0.062524$, correct to $5$ significant figures.</div></li><li><div class="list-item-paragraph">Find the probability that there are exactly $2$ unripe mangoes in $2$ boxes.</div></li></ol>
<p>A supermarket orders $50$ boxes of mangoes daily.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the probability that there are at least $9$ boxes with no unripe mango in a particular day.</div></li><li><div class="list-item-paragraph">When the supermarket places a daily order for&nbsp;$10$ weeks, estimate the probability that the mean number of boxes with no unripe mango in a day is between $3$ and $9$.</div></li></ol>

When a large number of mangoes are harvested, $37$% of them are unripe. The mangoes are packed randomly in boxes of $6$.

Show that the probability that there is no unripe mango in a box is $0.062524$, correct to $5$ significant figures.

Find the probability that there are exactly $2$ unripe mangoes in $2$ boxes.

A supermarket orders $50$ boxes of mangoes daily.

Find the probability that there are at least $9$ boxes with no unripe mango in a particular day.

When the supermarket places a daily order for $10$ weeks, estimate the probability that the mean number of boxes with no unripe mango in a day is between $3$ and $9$.

Binomial Probabilities Using Technology

Repeated Binomial Experiments

2015 TJC P2 Q9 [Modified]

Discrete Random Variables (DRV)

<p>Bernard is carrying out an experiment with a fair tetrahedral die, which has its four triangular faces numbered from '$6$' to '$9$', and a biased $10$-sided die numbered from '$1$' to '$10$'.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Bernard rolls the fair die $9$ times. Find the probability that the die shows a '$8$' between $3$ and $7$ times, inclusive.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Bernard now rolls the fair die $65$ times. Find the probability that the die shows a '$9$' more than $12$ times.</span></div></li></ol>
<p>The probability that the biased die shows a '$9$' is $\frac{1}{25}$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Bernard rolls the biased die $65$ times.</div><ol class="abc-list"><li><div class="list-item-paragraph"> Find the probability that the biased die shows a '$9$' more than $5$ times.</div></li><li><div class="list-item-paragraph">Bernard repeated the experiment 30 times. Estimate the probability that the average number of $9$' shown is at least $2.1$.</div></li></ol></li></ol>

Bernard is carrying out an experiment with a fair tetrahedral die, which has its four triangular faces numbered from '$6$' to '$9$', and a biased $10$-sided die numbered from '$1$' to '$10$'.

Bernard rolls the fair die $9$ times. Find the probability that the die shows a '$8$' between $3$ and $7$ times, inclusive.

Bernard now rolls the fair die $65$ times. Find the probability that the die shows a '$9$' more than $12$ times.

The probability that the biased die shows a '$9$' is $\frac{1}{25}$.

2015 IJC P2 Q7 [Modified]

<p>The amount of rainfall $X$ in a certain city per day has mean $1.3$ metres and variance $2.8$ square metres. The amount of rainfall for a sample of $60$ days is collected.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Calculate the mean and variance of $\overline{X}$.</div></li><li><div class="list-item-paragraph">Find the probability that the mean amount of rainfall exceeds $0.8$ metres.</div></li><li><div class="list-item-paragraph">State two assumptions about the sample used in obtaining this estimate in <strong>(ii)</strong>.</div></li></ol>

The amount of rainfall $X$ in a certain city per day has mean $1.3$ metres and variance $2.8$ square metres. The amount of rainfall for a sample of $60$ days is collected.

Assumptions for Sampling

<p>Suppose $X_{1},X_{2},...,X_{n}$ is a random sample of size $n$ taken from a population with mean $\mu$ and variance $\sigma^{2}$. If&nbsp;$\bar{X}$ is the sample mean, show that&nbsp;$\mathrm{E}(\bar{X})=\mu$ and $\mathrm{Var}(\bar{X})=\frac{\sigma^{2}}{n}$.</p>

Suppose $X_{1},X_{2},...,X_{n}$ is a random sample of size $n$ taken from a population with mean $\mu$ and variance $\sigma^{2}$. If $\bar{X}$ is the sample mean, show that $\mathrm{E}(\bar{X})=\mu$ and $\mathrm{Var}(\bar{X})=\frac{\sigma^{2}}{n}$.

Deriving Expectation and Variance of the Sample Mean

<p>Suppose ${{X}{1}},{{X}{2}},...,{{X}_{n}}$ is a random sample of size $n$ taken from a population with mean $\mu$ and variance ${{\sigma }^{2}}$. If  $\bar{X}$ is the sample mean, show that $\mathrm{E}\left( {\bar{X}} \right)=\mu $ and $\mathrm{Var}\left( {\bar{X}} \right)=\frac{{{\sigma }^{2}}}{n}$.</p>

Suppose ${{X}{1}},{{X}{2}},...,{{X}_{n}}$ is a random sample of size $n$ taken from a population with mean $\mu$ and variance ${{\sigma }^{2}}$. If  $\bar{X}$ is the sample mean, show that $\mathrm{E}\left( {\bar{X}} \right)=\mu $ and $\mathrm{Var}\left( {\bar{X}} \right)=\frac{{{\sigma }^{2}}}{n}$.

Worked Example 1

<p><strong>In this question you should state the parameters of any distributions you use.</strong></p>
<p>At a burger shop, the wait time, $W$ (in minutes), is defined to be the time from when a customer places an order at the counter until the food is collected. It was proposed that $W$ is modelled by $N(2,{{1.5}^{2}})$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Give a reason why this model is not suitable.</div><p style="text-align: right;">[1]</p></li></ol>
<p>A new model for $W$ is given by $N(5,{{1}^{2}})$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find the range of values of $k$ such that at least $90\%$ of customers experience a wait time longer than $k$ minutes.</div><p style="text-align: right;">[2]</p></li></ol>
<p>A customer bought burgers from the shop on three independent occasions, with wait times denoted by ${{W}_{1}}$, ${{W}_{2}}$ and ${{W}_{3}}$ respectively. Let $\bar{W}=\frac{{{W}_{1}}+{{W}_{2}}+{{W}_{3}}}{3}$.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Find the values of $\mathrm{Var}(\bar{W}-{{W}_{1}})$ and $\mathrm{Var}(\bar{W})+\mathrm{Var}({{W}_{1}})$ and hence, determine whether $\mathrm{Var}(\bar{W}-{{W}_{1}})=\mathrm{Var}(\bar{W})+\mathrm{Var}({{W}_{1}})$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the probability that the mean wait time is within one minute of the wait time on the first occasion.</div><p style="text-align: right;">[3]</p></li></ol>
<p>On the 4th occasion, the customer went to the restroom immediately after the order was placed. The time spent in the restroom, in $T$ minutes, is modelled by $T\sim N(7,{{1.5}^{2}})$.</p>
<p>Assume that the time taken to walk to the restroom is negligible and $W$ and $T$ follow independent normal distributions.</p>
<ol class="abc-list" start="5"><li><div class="list-item-paragraph">The customer is in a rush and will not wait for more than 3 minutes after leaving the restroom. Given that the burger is not ready for collection after the customer leaves the restroom, find the probability that the customer will leave the shop without collecting the burger.</div><p style="text-align: right;">[4]</p></li></ol>
<p>To reduce the wait time for customers, the shop later installs self-order kiosks. The wait time from when a customer places an order at the kiosk until the food is collected is modelled as being reduced by $20\%$ compared to that at the counter.</p>
<ol class="abc-list" start="6"><li><div class="list-item-paragraph">8 customers who ordered at the counter and 8 customers who ordered at the kiosk are randomly selected and their wait times are recorded. The wait time of each customer is independent of the others. Find the probability that exactly 2 of these 16 customers have a wait time of at least 5 minutes.</div><p style="text-align: right;">[4]</p></li></ol>

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

At a burger shop, the wait time, $W$ (in minutes), is defined to be the time from when a customer places an order at the counter until the food is collected. It was proposed that $W$ is modelled by $N(2,{{1.5}^{2}})$.

A new model for $W$ is given by $N(5,{{1}^{2}})$.

A customer bought burgers from the shop on three independent occasions, with wait times denoted by ${{W}_{1}}$, ${{W}_{2}}$ and ${{W}_{3}}$ respectively. Let $\bar{W}=\frac{{{W}_{1}}+{{W}_{2}}+{{W}_{3}}}{3}$.

On the 4th occasion, the customer went to the restroom immediately after the order was placed. The time spent in the restroom, in $T$ minutes, is modelled by $T\sim N(7,{{1.5}^{2}})$.

Assume that the time taken to walk to the restroom is negligible and $W$ and $T$ follow independent normal distributions.

To reduce the wait time for customers, the shop later installs self-order kiosks. The wait time from when a customer places an order at the kiosk until the food is collected is modelled as being reduced by $20\%$ compared to that at the counter.

Conditional Probability

Normal Probabilities and Quantiles

Probabilities Involving Multiple Binomial Variables

Linear Transformations of a Normal Variable

Linear Combinations of Independent Normal Variables

Assessing the Suitability of a Normal Model

2025 HCI P2 Q10

Probability

<p>A box contains two balls numbered $2$ and $k$ balls numbered $3$, where&nbsp;$k\ge 3$. In a game, three balls are drawn from the box at random and the score is obtained by multiplying together the numbers indicated on the balls that were drawn. Let the random variable $X$ denote the score obtained.</p>
<ol class="abc-list"><li data-list-item-id="item-1760847139471-5n3754fq8" data-list-item-path="item-1760847139471-5n3754fq8"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{E}\left( X \right)=\frac{9\left( 3{{k}^{2}}+3k+2 \right)}{\left( k+1 \right)\left( k+2 \right)}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760847140266-tshe7xjxf" data-list-item-path="item-1760847140266-tshe7xjxf"><div class="list-item-paragraph">Given that&nbsp;$\mathrm{E}\left( X \right)=\frac{144}{7}$, find the value of $k$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760847141833-1l2lm15gf" data-list-item-path="item-1760847141833-1l2lm15gf"><div class="list-item-paragraph">Assuming that each game is independent of one another, find the probability that the average score of $30$ games is more than $21$.<span class="inline-marks">[3]</span></div></li></ol>

A box contains two balls numbered $2$ and $k$ balls numbered $3$, where $k\ge 3$. In a game, three balls are drawn from the box at random and the score is obtained by multiplying together the numbers indicated on the balls that were drawn. Let the random variable $X$ denote the score obtained.

Constructing and Using Discrete Probability Distributions

Expectation and Variance of Discrete Random Variables

Determining Unknowns in Discrete Probability Distributions

Probability Using Permutations and Combinations

2024 YIJC P2 Q9

<ol class="abc-list"><li><div class="list-item-paragraph">The discrete random variable $X$ takes values $-1$, $0$, $1$ with probabilities $\frac{1}{4}$, $\frac{1}{2}$, $\frac{1}{4}$ respectively. The variable $\bar{X}$ is the mean of a random sample of $3$ values of $X$ (i.e. ${{X}_{1}}$, ${{X}_{2}}$ and ${{X}_{3}}$ are independent random variables).</div><div class="list-item-paragraph">Tabulate the probability distribution of $\bar{X}$, and use your values to calculate $\mathrm{Var}\left( {\bar{X}} \right)$.</div><div class="list-item-paragraph">Hence verify that $\mathrm{Var}\left( {\bar{X}} \right)=\frac{1}{3}\mathrm{Var}\left( X \right)$ in this case.</div></li></ol>

Linear Combinations of Random Variables

N1984 P2 Q11

<p>Two fair dice are thrown and the product of the two numbers on the top faces is computed. Let&nbsp;${{R}_{n}}$ be the remainder when the product is divided by $n$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the probability distribution of ${{R}_{2}}$ and hence find $\mathrm{E}\left( {{R}_{2}} \right)$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Show that&nbsp;$\mathrm{P}\left( {{R}_{6}}=0 \right)=\frac{5}{12}$.<span class="inline-marks">[1]</span></div></li></ol>
<p>The two dice are thrown m times and the product of the two numbers on the top faces are computed each time. $X$ is the number of products that are divisible by $6$ after $m$ throws.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Show that&nbsp;$\mathrm{P}\left( X=2 \right)=Am\left( m-1 \right){{\left( \frac{7}{12} \right)}^{m}}$, where&nbsp;$A$ is a constant to be determined. Given that x$\mathrm{P}\left( X=2 \right)=\mathrm{0}\mathrm{.03604}$, correct to $5$ decimal places, deduce the value of $m$, justifying your answer.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Class $C$ has&nbsp;$35$ students and class $D$ has&nbsp;$30$ students. Each student in the two classes throws the two dice $10$ times. Estimate the probability that the average number of products that are divisible by $6$ obtained by a student in class $C$ exceeds the average number of products that are divisible by $6$ obtained by a student in class $D$ by more than $0.2$.<span class="inline-marks">[5]</span></div></li></ol>

Two fair dice are thrown and the product of the two numbers on the top faces is computed. Let ${{R}_{n}}$ be the remainder when the product is divided by $n$.

The two dice are thrown m times and the product of the two numbers on the top faces are computed each time. $X$ is the number of products that are divisible by $6$ after $m$ throws.

Binomial Probabilities Using the Probability Mass Function

Comparing Independent Sample Means

2023 RI P2 Q10

<p>TGM adapted model: one round consists of exactly eight moves from $X$ on a square lattice. The diagram shows the part of the lattice containing $X$, $Y$ and $A$ to $E$.</p>
<p style="margin-left: 2rem">• The counter can only either move up or to the right along the grid lines.</p>
<p style="margin-left: 2rem">• At every movement, including the first from $X$, the player randomly moves the counter to the right with probability $p$ or up with probability $q$, $p+q=1$. The moves taken at each junction are independent.</p>
<p style="margin-left: 2rem">• A round is won if the counter passes through $Y$ and is at $B$ after its eighth move. Reaching the right-hand boundary earlier does not end the round; moves continue on the lattice. Every other final position is a loss.</p>
<div class="graph-block" 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data-width="407" data-height="139" data-proportional style="width:407px;height:139px;max-width:100%;display:flow-root;margin-left:auto;margin-right:auto;margin-top:var(--graph-gap, 0px);margin-bottom:calc(var(--graph-gap, 0px) + 0px);--graph-width:407px;--graph-aspect:407 / 139;"></div>
<ol class="abc-list"><li data-list-item-id="item-1760682801755-j7he753ry" data-list-item-path="item-1760682801755-j7he753ry"><div class="list-item-paragraph">Show that the probability of winning a round is $30{{p}^{5}}{{q}^{3}}$.<span class="inline-marks">[3]</span></div></li></ol>
<p>It is given that&nbsp;$p=\frac{4}{5}$. Wang plays one game which comprises $15$ rounds. Each round is independent of one another. Let the random variable $W$ be the number of rounds that Wang wins.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760682821261-yeqgynklh" data-list-item-path="item-1760682821261-yeqgynklh"><div class="list-item-paragraph">Find the probability that Wang wins at least one-third of the $15$ rounds.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760682821809-08cyine5a" data-list-item-path="item-1760682821809-08cyine5a"><div class="list-item-paragraph">Wang plays $40$ such games independently. Estimate the probability that the mean number of rounds won per game is at most one. Use the central limit theorem without a continuity correction.<span class="inline-marks">[3]</span></div></li></ol>
<p>TGM adaptation note: the original source does not specify what happens on first reaching the right-hand boundary. The eight-move rule above makes the retained $30p^5q^3$ calculation valid. The final central-limit-theorem task is optional extension beyond IB AA SL and HL.</p>

TGM adapted model: one round consists of exactly eight moves from $X$ on a square lattice. The diagram shows the part of the lattice containing $X$, $Y$ and $A$ to $E$.

• The counter can only either move up or to the right along the grid lines.

• At every movement, including the first from $X$, the player randomly moves the counter to the right with probability $p$ or up with probability $q$, $p+q=1$. The moves taken at each junction are independent.

• A round is won if the counter passes through $Y$ and is at $B$ after its eighth move. Reaching the right-hand boundary earlier does not end the round; moves continue on the lattice. Every other final position is a loss.

It is given that $p=\frac{4}{5}$. Wang plays one game which comprises $15$ rounds. Each round is independent of one another. Let the random variable $W$ be the number of rounds that Wang wins.

TGM adaptation note: the original source does not specify what happens on first reaching the right-hand boundary. The eight-move rule above makes the retained $30p^5q^3$ calculation valid. The final central-limit-theorem task is optional extension beyond IB AA SL and HL.

Probabilities of Successive Events

2023 DHS P2 Q8

<p>When a large number of mangoes are harvested, $37\%$ of them are unripe. The mangoes are packed randomly in boxes of six.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the probability that there is no unripe mango in a box is $0.062524$, correct to $5$ significant figures.</div></li><li><div class="list-item-paragraph">Find the probability that there are exactly $2$ unripe mangoes in $2$ boxes.</div></li></ol>
<p>A supermarket orders $50$ boxes of mangoes daily.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the probability that there are at least $9$ boxes with no unripe mango in a particular day.</div></li><li><div class="list-item-paragraph">When the supermarket places a daily order for $10$ weeks, estimate the probability that the mean number of boxes with no unripe mango in a day is between $3$ and $9$.</div></li></ol>

When a large number of mangoes are harvested, $37\%$ of them are unripe. The mangoes are packed randomly in boxes of six.

A supermarket orders $50$ boxes of mangoes daily.

<p>The owner of a fruit stall, Mr Lee, bought a large batch of durians from a durian farm.</p>
<p>The weights (in kg) of a sample of six durians from this batch are</p>
<p style="text-align: center">$1.0$,        $1.1$,       $1.3$,       $1.7$,       $a$,       $b$.</p>
<p>It is known that the unbiased estimates for the population mean and variance based on this sample are $1.25$kg and $0.095$kg$^{2}$ respectively.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the values of $a$ and $b$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Assuming that the weight of durians from Robert’s farm follows a normal distribution with mean $1.60$kg and standard deviation $0.3$kg, carry out an appropriate test at the $5%$ level to find out whether there is significant evidence that this batch of durians is not taken from Robert’s farm.</div><p style="text-align: right;">[4]</p></li></ol>
<p>The mean and standard deviation of the weight of papayas from Robert’s farm are $1.2$kg and $0.45$kg respectively. Mr Lee bought a batch of papayas from Robert’s farm. Find the least number of papayas in the batch such that the probability of the average weight exceeding $1.3$kg is less than $0.05$.</p><p style="text-align: right;">[3]</p>

The owner of a fruit stall, Mr Lee, bought a large batch of durians from a durian farm.

The weights (in kg) of a sample of six durians from this batch are

$1.0$,        $1.1$,       $1.3$,       $1.7$,       $a$,       $b$.

It is known that the unbiased estimates for the population mean and variance based on this sample are $1.25$kg and $0.095$kg$^{2}$ respectively.

The mean and standard deviation of the weight of papayas from Robert’s farm are $1.2$kg and $0.45$kg respectively. Mr Lee bought a batch of papayas from Robert’s farm. Find the least number of papayas in the batch such that the probability of the average weight exceeding $1.3$kg is less than $0.05$.

Unbiased Estimates of Population Parameters

Mean Tests for Normal Populations with Known Variance

Hypothesis Tests Using Critical Regions and $p$-Values

Two-Tailed Hypothesis Tests

2012 AJC MYCT P2 Q4

Hypothesis Testing

<ol class="i-list"><li><div class="list-item-paragraph">A multiple choice test consists of $20$ questions, each with four possible answers, of which only one is correct. If a student randomly chooses the answer to each question, find the probability of getting at least four but less than nine correct answers.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Suppose that each correct answer is awarded five marks and each incorrect answer carries a penalty of one mark, what is the expected score obtained by a student?</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Using a suitable approximation, find the probability that the mean score of $50$ randomly selected students is more than $12$ marks. State an assumption necessary for your answer to be valid.</div><p style="text-align: right;">[4]</p></li></ol>

Mean and Variance of a Binomial Distribution

NJC Sampling Tutorial Q4

<p>A company manufactures screen protectors for handphones. On average, $17\%$ of the screen protectors are cracked. The screen protectors are packaged into boxes of $20$. It should be assumed that the number of cracked screen protectors in a box of $20$ screen protectors follows a binomial distribution.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that the probability of having no more than $3$ cracked screen protectors in a randomly chosen box is $0.55041$, correct to $5$ significant figures.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Find the most probable number of cracked screen protectors in a randomly chosen box.<span class="inline-marks">[1]</span></div></li></ol>
<p>Every month, the company exports a large shipment of n cartons of screen protectors. In each carton, there are $8$ boxes of screen protectors. Every carton is assumed to be filled completely.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Using a suitable approximation, find the least value of $n$ such that in a randomly chosen month, there is at least a $99\%$ chance that the average number of boxes per carton having no more than $3$ cracked screen protectors is at most $5$.<span class="inline-marks">[4]</span></div></li></ol>
<p>To determine whether cartons are ready for local or overseas sales, cartons are randomly selected for inspection. In a carton,</p>
<p style="margin-left: 3rem">• if there are at least $4$ boxes that contain no more than $3$ cracked screen protectors each, the carton is ready for local sales;</p>
<p style="margin-left: 3rem">• if there are at least $6$ boxes that contain no more than $3$ cracked screen protectors each, the carton is ready for overseas sales.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">A carton that is not ready for overseas sales is randomly selected. Find the probability that the carton is ready for local sales.<span class="inline-marks">[3]</span></div></li></ol>

A company manufactures screen protectors for handphones. On average, $17\%$ of the screen protectors are cracked. The screen protectors are packaged into boxes of $20$. It should be assumed that the number of cracked screen protectors in a box of $20$ screen protectors follows a binomial distribution.

Every month, the company exports a large shipment of n cartons of screen protectors. In each carton, there are $8$ boxes of screen protectors. Every carton is assumed to be filled completely.

To determine whether cartons are ready for local or overseas sales, cartons are randomly selected for inspection. In a carton,

• if there are at least $4$ boxes that contain no more than $3$ cracked screen protectors each, the carton is ready for local sales;

• if there are at least $6$ boxes that contain no more than $3$ cracked screen protectors each, the carton is ready for overseas sales.

Mode of a Binomial Distribution

2022 EJC P2 Q6

<p>A computer can give independent observations of a random variable $X$ with probability distribution given by $\mathrm{P}\left( X=0 \right)=\frac{3}{4}$ and $\mathrm{P}\left( X=2 \right)=\frac{1}{4}$. It is programmed to output a value for the random variable $Y$ defined by $Y={{X}_{1}}+{{X}_{2}}$, where ${{X}_{1}}$ and ${{X}_{2}}$ are independent observations of $X$. Tabulate the probability distribution of $Y$, show that $E\left( Y \right)=1$.</p><p style="text-align: right;">[3]</p>
<p>The random variable $T$ is defined by $T={{Y}^{2}}$. Find $\mathrm{E}\left( T \right)$ and show that $\mathrm{Var}\left( T \right)=\frac{63}{4}$.</p><p style="text-align: right;">[3]</p>
<p>The computer is programmed to produce a large number $n$ of independent values of $T$ and to calculate the mean $M$ of these values. Find the smallest value of $n$ such that $\mathrm{P}\left( M<3 \right)>0.99$.</p><p style="text-align: right;">[4]</p>

A computer can give independent observations of a random variable $X$ with probability distribution given by $\mathrm{P}\left( X=0 \right)=\frac{3}{4}$ and $\mathrm{P}\left( X=2 \right)=\frac{1}{4}$. It is programmed to output a value for the random variable $Y$ defined by $Y={{X}_{1}}+{{X}_{2}}$, where ${{X}_{1}}$ and ${{X}_{2}}$ are independent observations of $X$. Tabulate the probability distribution of $Y$, show that $E\left( Y \right)=1$.

The random variable $T$ is defined by $T={{Y}^{2}}$. Find $\mathrm{E}\left( T \right)$ and show that $\mathrm{Var}\left( T \right)=\frac{63}{4}$.

The computer is programmed to produce a large number $n$ of independent values of $T$ and to calculate the mean $M$ of these values. Find the smallest value of $n$ such that $\mathrm{P}\left( M<3 \right)>0.99$.

N1998 P2 Q7

<p>On average, it is known that $1$ in $12$ peaches produced in a farm are classified as very sweet. A farmer randomly selected $40$ peaches from the farm.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, in this context, two necessary assumptions to model the random variable representing the number of very sweet peaches by a binomial distribution.</div><p style="text-align: right;">[2]</p></li></ol>
<p>The farmer randomly selects $5000$ peaches and packs them into boxes of $40$ peaches each for shipment.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Obtain an approximate distribution for the mean number of very sweet peaches in a box for this shipment. Hence find the probability that this mean number is at most $3$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">What is the least number of boxes that the farmer must ship so that the probability of obtaining not less than $200$ very sweet peaches is more than $0.99$ for this new shipment?</div><p style="text-align: right;">[4]</p></li></ol>

On average, it is known that $1$ in $12$ peaches produced in a farm are classified as very sweet. A farmer randomly selected $40$ peaches from the farm.

The farmer randomly selects $5000$ peaches and packs them into boxes of $40$ peaches each for shipment.

Minimum Number of Binomial Trials

Binomial Models and Assumptions

2022 ASRJC BT2 P2 Q6

<ol class="abc-list"><li><div class="list-item-paragraph">Patients who caught the common flu have a $98\%$ chance of full recovery. Find the probability that</div><ol class="i-list"><li><div class="list-item-paragraph">in a sample of $20$ patients, more than $2$ will recover fully,</div></li><li><div class="list-item-paragraph">In fifty sample of $20$ patients, the mean number of patients who will not recover is more than $0.5$.</div></li></ol></li><li><div class="list-item-paragraph">At the onset of a contagious flu virus, the probability that a patient survives is given by $p$. In a hospital ward of $50$ patients, past data shows that the probability of more than $48$ patients surviving the flu is $0.64$. Find the probability that a patient will survive the flu.</div></li></ol>

Determining an Unknown Binomial Success Probability

2009 NYJC P2

<p>On the average, $30\%$ of the students in Calculus Madness Institute (CMI) could do the Differential Equation question in Block Test Two. The principal randomly select a class of $30$ students to analyse the results.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, in the context of this question, two assumptions needed to model the results by a binomial distribution.</div></li><li><div class="list-item-paragraph">Find the probability that at least $6$ students in the class could do that question.</div></li><li><div class="list-item-paragraph">Find the probability that only $2$ students among the first $8$ selected students in the class could do the question given that at least $6$ students could do that question.</div></li><li><div class="list-item-paragraph">The probability of no more than $5$ students could do the question in a randomly selected class exceeds $0.9$. Find the largest possible number of students in that class.</div></li></ol>
<p>$50$ classes with $30$ students each are selected randomly. Find the probability that an average of at least $10$ students per class could do the question.</p>

On the average, $30\%$ of the students in Calculus Madness Institute (CMI) could do the Differential Equation question in Block Test Two. The principal randomly select a class of $30$ students to analyse the results.

$50$ classes with $30$ students each are selected randomly. Find the probability that an average of at least $10$ students per class could do the question.

Maximum Number of Binomial Trials

2012 SAJC P2 Q13

<p>In conjunction with the Great Singapore Sale, a certain electronics store is having a lucky draw for their customers. In each round of the lucky draw, the customer draws two balls randomly, one after another, with replacement from a box containing $20$ red balls, $30$ blue balls and $50$ white balls. The colour of each ball drawn is noted and points are awarded accordingly as follows.</p>
<table><thead><tr><th>Colour  of ball</th><th>Point(s)</th></tr></thead><tbody><tr><td>Red</td><td>5</td></tr><tr><td>Blue</td><td>4</td></tr><tr><td>White</td><td>1</td></tr></tbody></table>
<p>The customer’s score in each round of the lucky draw is the total number of points awarded for the balls drawn. To illustrate, a customer scores a total of $5$ points if he draws a blue ball and a white ball in a round of the lucky draw, regardless of the order of appearance of the balls.</p>
<p>You may assume that the $100$ balls are indistinguishable from each other apart from their colour.</p>
<p>Let $X$ denote the total number of points scored by a customer in a round of the lucky draw.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Tabulate the probability distribution of $X$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find $\mathrm{E}\left( X \right)$ and show that $\mathrm{Var}\left( X \right)=6.02$.</div><p style="text-align: right;">[2]</p></li></ol>
<p>Mr Lim participated in $50$ rounds of the lucky draw.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Using a suitable approximation, find the probability that Mr Lim’s average score is at least 6.</div><p style="text-align: right;">[3]</p></li></ol>
<p>A customer wins a cash voucher if his total score for one round of the lucky draw is more than $6$.</p>
<p>The total number of cash vouchers won by Mr Tan in $n$ rounds of the lucky draw is denoted by $Y$.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Find the least value of $n$ such that there is a probability of more than $0.7$ that Mr Tan will win more than $3$ cash vouchers in total.</div><p style="text-align: right;">[4]</p></li></ol>

In conjunction with the Great Singapore Sale, a certain electronics store is having a lucky draw for their customers. In each round of the lucky draw, the customer draws two balls randomly, one after another, with replacement from a box containing $20$ red balls, $30$ blue balls and $50$ white balls. The colour of each ball drawn is noted and points are awarded accordingly as follows.

The customer’s score in each round of the lucky draw is the total number of points awarded for the balls drawn. To illustrate, a customer scores a total of $5$ points if he draws a blue ball and a white ball in a round of the lucky draw, regardless of the order of appearance of the balls.

You may assume that the $100$ balls are indistinguishable from each other apart from their colour.

Let $X$ denote the total number of points scored by a customer in a round of the lucky draw.

Mr Lim participated in $50$ rounds of the lucky draw.

A customer wins a cash voucher if his total score for one round of the lucky draw is more than $6$.

The total number of cash vouchers won by Mr Tan in $n$ rounds of the lucky draw is denoted by $Y$.

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