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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

The masses of mangoes in kg have the distribution $\mathrm N(0.3,0.1^2)$.

Find the probability that two randomly selected mangoes weigh at least 0.5 kg in total.

Mark sells fruit boxes, each of which contain 10 randomly selected blueberries and 2 randomly selected mangoes. Let the number of unblemished blueberries in a randomly selected fruit box be $U$. On average, the probability of a blueberry being unblemished is $p$.

State a reason, in the context of the question, why $U$ might not be well-modelled by a binomial distribution.

Assume now that $U$ has the distribution $\mathrm B(10,p)$. The probability that a randomly selected fruit box contains at least 8 unblemished blueberries is 0.9.

Each fruit box is classified as grade A, B or C according to the following criteria:

Show that the probability that a randomly chosen fruit box is classified as grade C is 0.316, correct to 3 significant figures.

Hence find the probability that a randomly selected grade C fruit box contains at least 8 unblemished blueberries.

The selling prices of a grade A, grade B and grade C fruit box are $\$12$, $\$10$ and $\$6$ respectively. Find the probability that the total selling price of 5 randomly selected fruit boxes is less than $\$37$.

Mark decides to amend the criteria for grade A boxes as follows:

**Each** mango in the box weighs at least 0.3 kg **and** the box contains at least 9 unblemished blueberries.

State how the percentage of fruit boxes classified as grade A will change, and explain your answer.

Binomial Models and Assumptions

Symmetry and Properties of the Normal Distribution

Applying Probability Laws

Linear Combinations of Independent Normal Variables

Determining an Unknown Binomial Success Probability

2026 EJC P2 Q11

Probability

Discrete Random Variables (DRV)

Continuous Random Variables

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">The random variable $Y$ has a binomial distribution with mean $1.6$ and $\mathrm{P}(Y=0)=0.1296$. Find $\mathrm{P}(Y&gt;2)$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">A car park has a large number of parking lots. $80$ parking lots are observed and the number of occupied parking lots is denoted by $X$. State, in context, two assumptions needed for&nbsp;$X$ to be well modelled by a binomial distribution.</div></li></ol>
<p>The probability of a parking lot being occupied is $0.95$. Find the probability that out of the $80$ chosen parking lots, at least $90$% are occupied.</p>

The random variable $Y$ has a binomial distribution with mean $1.6$ and $\mathrm{P}(Y=0)=0.1296$. Find $\mathrm{P}(Y>2)$.

A car park has a large number of parking lots. $80$ parking lots are observed and the number of occupied parking lots is denoted by $X$. State, in context, two assumptions needed for $X$ to be well modelled by a binomial distribution.

The probability of a parking lot being occupied is $0.95$. Find the probability that out of the $80$ chosen parking lots, at least $90$% are occupied.

Binomial Probabilities Using Technology

Mean and Variance of a Binomial Distribution

2013 MJC P2 Q8 [Modified]

<p>National Fruit Company owns a large tomato farm. The tomatoes produced are harvested and sold in boxes of $25$. It is known that $100p\%$ of the tomatoes are rotten. For these boxes, the mean number of rotten tomatoes in a box is $1$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Explain why the context above may not be well-modelled by a binomial distribution.</div><p style="text-align: right;">[1]</p></li></ol>
<p>Assume now that the context above is well-modelled by a binomial distribution.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">State the value of $p$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the probability that a box chosen at random has less than $2$ rotten tomato.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">A customer chose a box and inspected the contents individually. Find the probability that the twenty-first tomato is the fourth rotten tomato and no rotten tomatoes are found subsequently.</div><p style="text-align: right;">[3]</p></li></ol>
<p>Boxes that contains at least $24$ tomatoes that are not rotten are deemed satisfactory.</p>
<ol class="i-list" start="5"><li><div class="list-item-paragraph">A customer first picks $3$ boxes of tomatoes, of which at least $2$ boxes are satisfactory. The customer then decides to buy another $5$ boxes. Find the probability that exactly $6$ of the $8$ boxes are satisfactory.</div><p style="text-align: right;">[4]</p></li></ol>

National Fruit Company owns a large tomato farm. The tomatoes produced are harvested and sold in boxes of $25$. It is known that $100p\%$ of the tomatoes are rotten. For these boxes, the mean number of rotten tomatoes in a box is $1$.

Assume now that the context above is well-modelled by a binomial distribution.

Boxes that contains at least $24$ tomatoes that are not rotten are deemed satisfactory.

Conditional Probability

Probabilities of Successive Events

Binomial Probabilities Using the Probability Mass Function

Repeated Binomial Experiments

Waiting-Time and Stopping-Rule Probabilities

2021 NJC P2 Q8

<p>A company sells toys in blind boxes. Each blind box contains exactly one toy. On average, $2\%$ of such blind boxes contain a faulty toy.</p>
<p>A collector buys $15$ such blind boxes at random.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, in context, two assumptions needed for the number of faulty toys he gets to be well modelled by a binomial distribution.</div><p style="text-align: right;">[2]</p></li></ol>
<p>Assume now that the number of faulty toys the collector gets has a binomial distribution.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the probability that the collector gets no less than $3$ faulty toys.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the probability that the fourteenth box is the second box with a faulty toy.</div><p style="text-align: right;">[2]</p></li></ol>
<p>The company also sells keychains. The probability of a keychain being faulty is $p$. The number of faulty keychains follows a binomial distribution. Faults on keychains are independent of faults on toys in blind boxes.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Write down an expression, in terms of $p$, for the probability that in a random sample of $3$ keychains, exactly one is faulty.</div><p style="text-align: right;">[1]</p></li></ol>
<p>A surprise pack contains $2$ randomly chosen blind boxes and $3$ randomly chosen keychains.</p>
<p>Given that a randomly selected surprise pack contains at most $1$ faulty item, the probability that one of the toys is the only faulty item in the pack is $0.035$.</p>
<ol class="i-list" start="5"><li><div class="list-item-paragraph">Write down an equation satisfied by $p$. Hence, find the value of $p$.</div><p style="text-align: right;">[4]</p></li></ol>

A company sells toys in blind boxes. Each blind box contains exactly one toy. On average, $2\%$ of such blind boxes contain a faulty toy.

A collector buys $15$ such blind boxes at random.

Assume now that the number of faulty toys the collector gets has a binomial distribution.

The company also sells keychains. The probability of a keychain being faulty is $p$. The number of faulty keychains follows a binomial distribution. Faults on keychains are independent of faults on toys in blind boxes.

A surprise pack contains $2$ randomly chosen blind boxes and $3$ randomly chosen keychains.

Given that a randomly selected surprise pack contains at most $1$ faulty item, the probability that one of the toys is the only faulty item in the pack is $0.035$.

Probabilities Involving Multiple Binomial Variables

2025 SAJC P2 Q8

<p>Always Cool Company produces reusable ice pack gels. It is known that, on average, $p\%$ of the reusable ice pack gels leak. Each day the quality control department takes a random sample of $120$ reusable ice pack gels and checks them for leakage. The number of reusable ice pack gels that leak in a sample is denoted by the random variable $X$.</p>
<ol class="i-list"><li data-list-item-id="item-1760688228763-uzpaivtwb" data-list-item-path="item-1760688228763-uzpaivtwb"><div class="list-item-paragraph">State, in the context of the question, two assumptions needed for&nbsp;$X$ to be well modelled by a binomial distribution.<span class="inline-marks">[2]</span></div></li></ol>
<p>Assume now that&nbsp;$X$ can be modelled by a binomial distribution,</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1760688247723-lbq7o4hrc" data-list-item-path="item-1760688247723-lbq7o4hrc"><div class="list-item-paragraph">It is given that the most likely number of reusable ice pack gels that leak in a sample is $3$. Find the range of values of $p$.<span class="inline-marks">[3]</span></div></li></ol>
<p><strong>For the rest of the question, the value of $p$ is taken to be $3$.</strong></p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760688271407-x2zz1jj7g" data-list-item-path="item-1760688271407-x2zz1jj7g"><div class="list-item-paragraph">Find the probability that the quality control department finds at most $4$ reusable ice pack gels that leaks in a sample.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760688273411-s4xldhdti" data-list-item-path="item-1760688273411-s4xldhdti"><div class="list-item-paragraph">If the quality control department found at least $5$ reusable ice pack gels that leak in a sample, find the probability that there are fewer than $9$ ice pack gels that leak in a sample.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760688281300-ih9acns3w" data-list-item-path="item-1760688281300-ih9acns3w"><div class="list-item-paragraph">The number of reusable ice pack gels that leak in a sample is independent of that of other samples. Find the probability that, in $10$ randomly chosen samples, the eighth sample is the third sample with at most $4$ reusable ice pack gels that leak.<span class="inline-marks">[3]</span></div></li></ol>

Always Cool Company produces reusable ice pack gels. It is known that, on average, $p\%$ of the reusable ice pack gels leak. Each day the quality control department takes a random sample of $120$ reusable ice pack gels and checks them for leakage. The number of reusable ice pack gels that leak in a sample is denoted by the random variable $X$.

Assume now that $X$ can be modelled by a binomial distribution,

**For the rest of the question, the value of $p$ is taken to be $3$.**

Mode of a Binomial Distribution

2020 MI P2 Q9

<p>The table shows the probability distribution of a discrete random variable $Y$.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 600px;"><colgroup><col style="width: 100px;"><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: center; --cell-justify: center">$y$</th><th style="text-align: center; --cell-justify: center">$-1$</th><th style="text-align: center; --cell-justify: center">$0$</th><th style="text-align: center; --cell-justify: center">$1$</th><th style="text-align: center; --cell-justify: center">$2$</th><th style="text-align: center; --cell-justify: center">$3$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$\mathrm{P}(Y = y)$</td><td style="text-align: center; --cell-justify: center">$\frac{32}{81}$</td><td style="text-align: center; --cell-justify: center">$a$</td><td style="text-align: center; --cell-justify: center">$\frac{8}{27}$</td><td style="text-align: center; --cell-justify: center">$b$</td><td style="text-align: center; --cell-justify: center">$c$</td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-paragraph">Given that&nbsp;$\mathrm{E}\left( \left| Y-3 \right| \right)=\frac{232}{81}$, show that&nbsp;$2b+3c=\frac{19}{81}$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Given further that&nbsp;$\mathrm{Var}\left( Y \right)=\frac{7736}{6561}$, find the values of $a$, $b$ and $c$ in rational form.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Justify if&nbsp;$Y$ follows a binomial distribution.<span class="inline-marks">[1]</span></div></li></ol>

The table shows the probability distribution of a discrete random variable $Y$.

Expectation and Variance of Discrete Random Variables

2022 NJC P2 Q8

<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

Central Limit Theorem

Sample Mean and Its Sampling Distribution

2022 ASRJC BT2 P2 Q6

Sampling

<p>A company manufactures a wide variety of components for use in domestic appliances.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">The company introduces a new type of light fitting for refrigerators. Each day a supervisor selects a sample of $100$ of the light fittings for testing.</div><ol class="i-list"><li><div class="list-item-paragraph">How should the light fittings be selected? Give a reason for this method of selection.</div><p style="text-align: right;">[2]</p><div class="list-item-paragraph">The supervisor records the number of faulty light fittings found on each of the $150$ working days.</div><div class="list-item-paragraph">Her results are shown in the table.</div><div class="list-item-image"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760338294891-ydesdmm9oj.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAzMzgyOTQ4OTEteWRlc2RtbTlvai5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTI3OTQzLCJleHAiOjE3OTM1MjgyNDN9.ShYOVupsTutKNnsNP3_O1UAfcaL3zofRjefeN9yOToQ" alt="Sub-item image" /></div></li><li><div class="list-item-paragraph">Use the information in the table to estimate $p$, the probability that a light fitting is faulty.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Assuming that the number of faulty fittings found each day follows the binomial distribution $\mathrm{B}\left( 100,\mathrm{ }p \right)$, find the expected number of days on which $3$ faulty fittings are found in a period of $150$ days.</div><p style="text-align: right;">[2]</p></li></ol></li><li><div class="list-item-paragraph">The company also makes heating elements for electric ovens. A fixed number of randomly chosen heating elements are tested each day and the number found to be faulty is denoted by $X$.</div><ol class="i-list"><li><div class="list-item-paragraph">State, in context, two assumptions needed for $X$ to be well modelled by a binomial distribution.</div><p style="text-align: right;">[2]</p><div class="list-item-paragraph">Assume now that $X$ has the distribution $\mathrm{B}\left( 80,\mathrm{ }0.02 \right)$.</div></li><li><div class="list-item-paragraph">Find the probability that, on a randomly chosen day, the number of elements found to be faulty is between $1$ and $4$ inclusive.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the probability that, in a randomly chosen $5$-day working week, more than $3$ elements are found to be faulty on at least $2$ days.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the probability that, in a randomly chosen $5$-day working week, no more than $8$ faulty elements are found in total.</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

A company manufactures a wide variety of components for use in domestic appliances.

Population and Random Samples

N2021 P2 Q11

<p>A sample of $6$ objects is to be drawn from a population in which $20%$ are defective. Explain why, if there are $10\,000$ objects in the whole population, the probability of obtaining $3$ defective objects in the sample can be very accurately estimated by using the binomial probability formula, whereas, if there are only $20$ objects in the population, this method gives quite the wrong answer.</p>
<p>For the large population, use this method to estimate the probability that a sample of $6$ contains $3$ or fewer defective objects.</p>
<p style="text-align: right">[ $0.983 $]</p>
<p>Another large population contains a proportion $p$ of defective items. Write down an expression in terms of $p$ for $P$, the probability that a sample of $6$ contains exactly $2$ defectives. Find $\frac{\mathrm{d}P}{\mathrm{d}p}$ and show that $P$ is greatest when $p=\frac{1}{3}$.</p>

A sample of $6$ objects is to be drawn from a population in which $20%$ are defective. Explain why, if there are $10\,000$ objects in the whole population, the probability of obtaining $3$ defective objects in the sample can be very accurately estimated by using the binomial probability formula, whereas, if there are only $20$ objects in the population, this method gives quite the wrong answer.

For the large population, use this method to estimate the probability that a sample of $6$ contains $3$ or fewer defective objects.

Another large population contains a proportion $p$ of defective items. Write down an expression in terms of $p$ for $P$, the probability that a sample of $6$ contains exactly $2$ defectives. Find $\frac{\mathrm{d}P}{\mathrm{d}p}$ and show that $P$ is greatest when $p=\frac{1}{3}$.

Product Rule

Problems Involving Maximisation and Minimisation

Chain Rule

NYJC Tutorial Q2

Differentiation

Applications of Differentiation

Techniques of Differentiation

<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>Based on past statistical data of high speed train journey, there is a $8\%$ chance that a passenger with reservation made will not show up. In order to maximise revenue, a high speed train company accepts more reservations than the passenger capacity of its trains.</p>
<ol class="i-list"><li data-list-item-id="item-1760260797041-fl6nn7gbg" data-list-item-path="item-1760260797041-fl6nn7gbg"><div class="list-item-paragraph">State $2$ assumptions needed such that the number of passengers who do not show up for a high speed train journey may be well modelled by a Binomial distribution.<span class="inline-marks">[2]</span></div></li></ol>
<p>A train company operates a high speed train from Burong Lake to Huahin Cove which has a capacity of $280$ passengers.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1760260811831-77gnmqp97" data-list-item-path="item-1760260811831-77gnmqp97"><div class="list-item-paragraph">Find the probability that when&nbsp;$300$ reservations are accepted, the high speed train journey is overbooked, i.e. there is not enough seats available for the passengers who show up.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760260813009-pm44nye4r" data-list-item-path="item-1760260813009-pm44nye4r"><div class="list-item-paragraph">Find the maximum number of reservations that should be accepted in order to ensure that the probability of overbooking is less than $1\%$.<span class="inline-marks">[3]</span></div></li></ol>
<p>The high speed train operates once daily throughout the year and $300$ reservations are accepted for each journey.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1760260842185-3ymwexvh4" data-list-item-path="item-1760260842185-3ymwexvh4"><div class="list-item-paragraph">Find the probability that no high speed train journey is overbooked in a week.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760260842875-ed4rv7k7m" data-list-item-path="item-1760260842875-ed4rv7k7m"><div class="list-item-paragraph">Two such journeys from Burong Lake to Huahin Cove with $300$ reservations each are examined. If the total number of passengers who turned up for these two journeys is at most $550$ passengers, find the probability that one of the journeys is overbooked by at most $2$ passengers.<span class="inline-marks">[3]</span></div></li></ol>

Based on past statistical data of high speed train journey, there is a $8\%$ chance that a passenger with reservation made will not show up. In order to maximise revenue, a high speed train company accepts more reservations than the passenger capacity of its trains.

A train company operates a high speed train from Burong Lake to Huahin Cove which has a capacity of $280$ passengers.

The high speed train operates once daily throughout the year and $300$ reservations are accepted for each journey.

2018 SRJC P2 Q12

<p>A variety of fish are bred in large batches with $20\%$ is the fish bred being salmon fish.</p>
<ol class="i-list"><li data-list-item-id="item-1760259446910-7kvo1vlfl" data-list-item-path="item-1760259446910-7kvo1vlfl"><div class="list-item-paragraph">Explain why a binomial distribution is appropriate for modelling the number of salmon fish in a catch of $n$ fish.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760259448027-emlz3aznl" data-list-item-path="item-1760259448027-emlz3aznl"><div class="list-item-paragraph">What is the probability that the ${{10}^{\mathrm{th}}}$ fish caught is the sixth salmon fish?<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760259448733-jp1jlbeuo" data-list-item-path="item-1760259448733-jp1jlbeuo"><div class="list-item-paragraph">A large number, $n$, of fish is caught.</div><ol class="abc-list"><li data-list-item-id="subitem-1760259485723-a8ey9ajzm" data-list-item-path="item-1760259448733-jp1jlbeuo/subitem-1760259485723-a8ey9ajzm"><div class="list-item-paragraph">If the probability of having $10$ or fewer salmon fish caught is at most $0.0227$, find the least possible number of fish caught.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="subitem-1760259486540-s2kulexwg" data-list-item-path="item-1760259448733-jp1jlbeuo/subitem-1760259486540-s2kulexwg"><div class="list-item-paragraph">Find the possible values of $n$ such that&nbsp;$15$ is the most likely number of salmon fish caught.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

A variety of fish are bred in large batches with $20\%$ is the fish bred being salmon fish.

2010 ACJC P2 Q12 [Modified]

<p>To aid in contact tracing during the Covid-19 pandemic, the government of a country introduces the use of TraceWith App and Token for its people. Each individual in the population chooses to use either the App or the Token. On average, $p\%$ of the population chose to use the TraceWith Token.</p>
<p>$n$ individuals are randomly selected for a survey and the number of individuals who use the TraceWith Token is denoted by the random variable $X$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State in context, two assumptions needed for $X$ to be well modelled by a binomial distribution.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Given that $X$ is modelled by a binomial distribution, $\mathrm{E}\left( X \right)=3.125$ and $\mathrm{Var}\left( X \right)=\mathrm{2}\mathrm{.734375}$, determine the values of $n$ and $p$.</div><p style="text-align: right;">[3]</p></li></ol>
<p>Assume that $X$ has a binomial distribution $\mathrm{B}\left( 35,0.2 \right)$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Determine the most likely number of individuals who use the TraceWith Token.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">The probability that $a$ or more individuals use the TraceWith Token is more than $0.9$. Find the largest value of $a$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Given that fewer than half of the surveyed individuals use the TraceWith Token, find the probability that more than $10$ surveyed individuals use the token.</div><p style="text-align: right;">[3]</p></li></ol>

To aid in contact tracing during the Covid-19 pandemic, the government of a country introduces the use of TraceWith App and Token for its people. Each individual in the population chooses to use either the App or the Token. On average, $p\%$ of the population chose to use the TraceWith Token.

$n$ individuals are randomly selected for a survey and the number of individuals who use the TraceWith Token is denoted by the random variable $X$.

Assume that $X$ has a binomial distribution $\mathrm{B}\left( 35,0.2 \right)$.

2022 RVHS J2 CT P1 Q11

<p>A trial for a new treatment for a particular disease is to be carried out. A sample of $n$ patients suffering from this disease is taken and these patients are invited to take part in the trial. The number of patients in the sample who accept the invitation is denoted by $X$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, in context, two assumptions needed for $X$ to be well modelled by a binomial distribution.</div><p style="text-align: right;">[2]</p></li></ol>
<p>Assume now that $X$ has the distribution $\mathrm{B}(10,\,0.25)$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the most probable number of patients who accept the invitation.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the probability that exactly $6$ patients accept the invitation given that at least $1$ patient accepts the invitation.</div><p style="text-align: right;">[2]</p></li></ol>

A trial for a new treatment for a particular disease is to be carried out. A sample of $n$ patients suffering from this disease is taken and these patients are invited to take part in the trial. The number of patients in the sample who accept the invitation is denoted by $X$.

Assume now that $X$ has the distribution $\mathrm{B}(10,\,0.25)$.

2017 VJC J2 MYE P1 Q11

<p>It is a common practice for airlines to sell more plane tickets than the number of seats available. This is to maximise their profits as it is expected that some passengers will not turn up for the flight. The plane used by Victoria Airline for her daily 10 am flight from Singapore to Hong Kong has a maximum capacity of $150$ seats. For this particular flight, $154$ tickets are sold every day.</p>
<p>On average, $p$ out of $100$ customers who have purchased a plane ticket for this flight turn up. Customers who turn up after the flight is full will be turned away. The number of customers who turn up for the 10 am flight, on a randomly chosen day, is denoted by $X$.</p>
<ol class="i-list"><li data-list-item-id="item-1760153032496-2yuham59c" data-list-item-path="item-1760153032496-2yuham59c"><div class="list-item-paragraph">State, in the context of this question, two assumptions needed to model $X$ by a binomial distribution.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760153033073-s5xfbg9l1" data-list-item-path="item-1760153033073-s5xfbg9l1"><div class="list-item-paragraph">Explain why one of the assumptions stated in part <strong>(i)</strong> may not hold in this context.</div></li></ol>
<p>Assume now that these assumptions do in fact hold.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760153096028-gfsze2kyz" data-list-item-path="item-1760153096028-gfsze2kyz"><div class="list-item-paragraph">It is known that there is a $0.05$ probability that at least $153$ customers will turn up for the 10 am flight. Write down an equation for the value of $p$, and find this value numerically.<span class="inline-marks">[3]</span></div></li></ol>
<p>It is given instead that&nbsp;$p=94$.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1760153119717-82mtj5wkq" data-list-item-path="item-1760153119717-82mtj5wkq"><div class="list-item-paragraph">Find the probability that, on a randomly chosen day,</div><ol class="abc-list"><li data-list-item-id="subitem-1760153139805-p36i05ylo" data-list-item-path="item-1760153119717-82mtj5wkq/subitem-1760153139805-p36i05ylo"><div class="list-item-paragraph">there are at least $141$ but not more than $148$ customers who turn up for the 10 am flight,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760153140644-akest2zqr" data-list-item-path="item-1760153119717-82mtj5wkq/subitem-1760153140644-akest2zqr"><div class="list-item-paragraph">every customer who turns up gets a seat on the 10 am flight.<span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="item-1760153221074-nk4tey9jg" data-list-item-path="item-1760153221074-nk4tey9jg"><div class="list-item-paragraph">Find the probability that every customer who turns up gets a seat on the 10 am flight on more than $5$ days in a week.<span class="inline-marks">[3]</span></div></li></ol>

It is a common practice for airlines to sell more plane tickets than the number of seats available. This is to maximise their profits as it is expected that some passengers will not turn up for the flight. The plane used by Victoria Airline for her daily 10 am flight from Singapore to Hong Kong has a maximum capacity of $150$ seats. For this particular flight, $154$ tickets are sold every day.

On average, $p$ out of $100$ customers who have purchased a plane ticket for this flight turn up. Customers who turn up after the flight is full will be turned away. The number of customers who turn up for the 10 am flight, on a randomly chosen day, is denoted by $X$.

Assume now that these assumptions do in fact hold.

2017 VJC P2 Q10

<p>A factory manufactures a large number of erasers in a variety of colours. Each box of erasers contains $36$ randomly chosen erasers. On average, $20$% of erasers in the box are blue.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, in context, two assumptions needed for the number of blue erasers in a box to be well modelled by a binomial distribution.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the probability that a randomly chosen box of erasers contain at most six blue erasers.</div><p style="text-align: right;">[1]</p></li></ol>
<p>$200$ randomly chosen boxes are packed into a carton. A carton is considered acceptable if at least $40$% of the boxes contain at most six blue erasers each.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the probability that a randomly chosen carton is acceptable.</div><p style="text-align: right;">[3]</p></li></ol>
<p>The cartons are exported by sea. Over a one-year period, there are $30$ shipments of $150$ cartons each.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Using a suitable approximation, find the probability that the mean number of acceptable cartons per shipment for the year is less than $80$.</div></li></ol>
<p>The owner decided to change the proportion of blue erasers to $p$. A box of erasers is chosen.</p>
<ol class="i-list" start="5"><li><div class="list-item-paragraph">Write down in terms of $p$, the probability that the box contains exactly one blue eraser.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">The probability that a box contains exactly one blue eraser is twice the probability that the box contains exactly two blue erasers. Write an equation in terms of $p$, and hence find the value of $p$.</div><p style="text-align: right;">[2]</p></li></ol>

A factory manufactures a large number of erasers in a variety of colours. Each box of erasers contains $36$ randomly chosen erasers. On average, $20$% of erasers in the box are blue.

$200$ randomly chosen boxes are packed into a carton. A carton is considered acceptable if at least $40$% of the boxes contain at most six blue erasers each.

The cartons are exported by sea. Over a one-year period, there are $30$ shipments of $150$ cartons each.

The owner decided to change the proportion of blue erasers to $p$. A box of erasers is chosen.

2019 SAJC P2 Q10

<p>Factory $A$ manufactures a large batch of light bulbs. It is known that on average, $1$ out of $200$ light bulbs manufactured by Factory $A$, is defective. A random sample of $180$ light bulbs is inspected. The batch is accepted if the sample contains less than $r$ defective light bulbs.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Explain why the context above may not be well-modelled by a binomial distribution.</div><p style="text-align: right;">[1]</p></li></ol>
<p>Assume now that the context above is well-modelled by a binomial distribution.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Determine the value of $r_{{}}^{{}}$ such that the probability of accepting the batch is $0.998$.</div><p style="text-align: right;">[1]</p></li></ol>
<p>In Factory $B$, a random sample of $30$ light bulbs is taken from a large batch. If the sample contains no defective light bulbs, the batch is accepted. The batch is rejected if the sample contains more than two defective light bulbs. If the sample contains one or two defective light bulbs, a second random sample of $30$ light bulbs is chosen and the batch is accepted only if this second sample contains no defectives. It is known that Factory $B$ produces $(100p)%$ defective light bulbs.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the probability that the batch is accepted. Leave your answer in terms of $p$.</div><p style="text-align: right;">[3]</p></li></ol>
<p>Forty random samples of $30$ light bulbs are taken from each of the two factories $A$ and $B$.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Given that $p=0.007$ and there is exactly one defective bulb, find the probability that it is from Factory $B$.</div><p style="text-align: right;">[4]</p></li></ol>

Factory $A$ manufactures a large batch of light bulbs. It is known that on average, $1$ out of $200$ light bulbs manufactured by Factory $A$, is defective. A random sample of $180$ light bulbs is inspected. The batch is accepted if the sample contains less than $r$ defective light bulbs.

In Factory $B$, a random sample of $30$ light bulbs is taken from a large batch. If the sample contains no defective light bulbs, the batch is accepted. The batch is rejected if the sample contains more than two defective light bulbs. If the sample contains one or two defective light bulbs, a second random sample of $30$ light bulbs is chosen and the batch is accepted only if this second sample contains no defectives. It is known that Factory $B$ produces $(100p)%$ defective light bulbs.

Forty random samples of $30$ light bulbs are taken from each of the two factories $A$ and $B$.

Law of Total Probability

Multistage Binomial Sampling

2017 NJC P2 Q10

<p>In an opinion poll before an election, a sample of $30$ voters is obtained.</p>
<ol class="i-list"><li><div class="list-item-paragraph">The number of voters in the sample who support the Alliance Party is denoted by $A$.</div><div class="list-item-paragraph">State, in context, what must be assumed for $A$ to be well modelled by a binomial distribution.</div><p style="text-align: right;">[2]</p></li></ol>
<p>Assume now that $A$ has the distribution $\mathrm{B}\left( 30,p \right)$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Given that $p=0.15$, find $\mathrm{P}\left( A=3\,\,\mathrm{or}\,\,4 \right)$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">For an unknown value of $p$, it is given that $\mathrm{P}\left( A=15 \right)=0.06864$ correct to $5$ decimal places. Show that $p$ satisfies an equation of the form $p\left( 1-p \right)=k$, where $k$ is a constant to be determined. Hence find the value of $p$ to a suitable degree of accuracy. Given that $p<0.5$.</div><p style="text-align: right;">[3]</p></li></ol>

In an opinion poll before an election, a sample of $30$ voters is obtained.

Assume now that $A$ has the distribution $\mathrm{B}\left( 30,p \right)$.

N2012 P2 Q9 [Modified]

<p>A female doctor wishes to test a new treatment for a certain genetic disease. On a particular day, she asks $n$ (where $n$ is fixed) patients who come to her clinic suffering from this disease if they are willing to take part in a trial. Based on past experiences, she finds that the probability of such a patient willing to take part is $0.3$. The number of patients who are willing to take part is the random variable $R$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, in the context of this question, two assumptions needed to model $R$ by a binomial distribution.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Explain why one of the assumptions stated in part <strong>(i)</strong> may not hold in this context.</div><p style="text-align: right;">[1]</p></li></ol>
<p>Assume now that these assumptions do in fact hold.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Given that $n=8$, find the probability that $R$ is at least $2$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">If the chance of at least $2$ patients willing to take part in this trial is more than $90\%$, find the least possible integer value of $n$.</div><p style="text-align: right;">[3]</p></li></ol>

A female doctor wishes to test a new treatment for a certain genetic disease. On a particular day, she asks $n$ (where $n$ is fixed) patients who come to her clinic suffering from this disease if they are willing to take part in a trial. Based on past experiences, she finds that the probability of such a patient willing to take part is $0.3$. The number of patients who are willing to take part is the random variable $R$.

2014 PJC P2 Q9

<p>On average $8\%$ of a certain brand of kitchen lights are faulty. The lights are sold in boxes of $12$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, in context, two assumptions needed for the number of faulty lights in a box to be well modelled by a binomial distribution.</div><p style="text-align: right;">[2]</p></li></ol>
<p>Assume now that the number of faulty lights in a box has a binomial distribution.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the probability that a box of $12$ of these kitchen lights contains at least $1$ faulty lights.</div><p style="text-align: right;">[1]</p></li></ol>
<p>The boxes are packed into cartons. Each carton contains $20$ boxes.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the probability that each box in one randomly selected carton contains at least one faulty light.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the probability that there are at least $20$ faulty lights in a randomly selected carton.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Explain why the answer to part <strong>(iv)</strong> is greater than the answer to part <strong>(iii)</strong>.</div><p style="text-align: right;">[1]</p></li></ol>
<p>The manufacturer introduces a quick test to check if lights are faulty. Lights identified as faulty are discarded. If a light is faulty there is a $95\%$ chance that the quick test will correctly identify the light as faulty. If the light is not faulty, there is a $6\%$ chance that the quick test will incorrectly identify the light as faulty.</p>
<ol class="i-list" start="6"><li><div class="list-item-paragraph">Find the probability that a light identified as faulty by the quick test is not faulty.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the probability that the quick test correctly identifies lights as faulty or not faulty.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Discuss briefly whether the quick test is worthwhile.</div><p style="text-align: right;">[1]</p></li></ol>

On average $8\%$ of a certain brand of kitchen lights are faulty. The lights are sold in boxes of $12$.

Assume now that the number of faulty lights in a box has a binomial distribution.

The boxes are packed into cartons. Each carton contains $20$ boxes.

The manufacturer introduces a quick test to check if lights are faulty. Lights identified as faulty are discarded. If a light is faulty there is a $95\%$ chance that the quick test will correctly identify the light as faulty. If the light is not faulty, there is a $6\%$ chance that the quick test will incorrectly identify the light as faulty.

Math Question Bank