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E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

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IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<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 $9$.

Central Limit Theorem

Binomial Probabilities Using Technology

Repeated Binomial Experiments

Sample Mean and Its Sampling Distribution

2015 TJC P2 Q9 [Modified]

Sampling

Discrete Random Variables (DRV)

<p>In a large population, $12\%$ of the population is left handed. Samples of $7$ people are randomly chosen from the population.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Calculate the probability that more than $2$ people in a sample are left handed.</div></li><li><div class="list-item-paragraph">$3$ such samples of $7$ people are taken. Find the probability that</div><ol class="i-list"><li><div class="list-item-paragraph">Each of the $3$ samples has more than $2$ people who are left-handed.</div></li><li><div class="list-item-paragraph">One of the $3$ samples has exactly $1$ person who is left-handed, while the remaining $2$ samples have more than $2$ people who are left handed.</div></li><li><div class="list-item-paragraph">One of the $3$ samples has less than $3$ people who are left-handed, another has exactly $3$ people who are left-handed and the remaining sample has more than $3$ people who are left-handed.</div></li><li><div class="list-item-paragraph">There are total of $3$ people who are left-handed from the $3$ samples.</div></li></ol></li></ol>

In a large population, $12\%$ of the population is left handed. Samples of $7$ people are randomly chosen from the population.

Probabilities Involving Multiple Binomial Variables

<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

Binomial Models and Assumptions

Probabilities of Successive Events

Binomial Probabilities Using the Probability Mass Function

Determining an Unknown Binomial Success Probability

Mean and Variance of a Binomial Distribution

Waiting-Time and Stopping-Rule Probabilities

2021 NJC P2 Q8

Probability

<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>In a calculators manufacturing factory, it is found that&nbsp;$1.5\%$ of the calculators produced are defective.</p>
<p>In a random sample of $90$ calculators, find the probability that</p>
<ol class="i-list"><li><div class="list-item-paragraph">there are exactly $2$ defective calculators,<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">the ${{90}^{\mathrm{th}}}$ calculator is the second defective calculator given that there are exactly $2$ defective calculators in that sample.<span class="inline-marks">[3]</span></div></li></ol>
<p>The calculators are packed in boxes of $90$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the probability that not more than $1$ box of calculators, out of $60$ boxes, contain more than $2$ defective calculators.<span class="inline-marks">[5]</span></div></li></ol>

In a calculators manufacturing factory, it is found that $1.5\%$ of the calculators produced are defective.

In a random sample of $90$ calculators, find the probability that

The calculators are packed in boxes of $90$.

2015 SRJC P2 Q9

<p>On average, $7.8\%$ of markers produced by a factory are faulty. Each day, the factory’s quality manager picks a random sample of n markers and inspects the number of faulty markers found in the sample. The number of faulty markers found in the sample is the random variable $X$ and whether the markers are faulty are independent of one another.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Explain what is meant by a random sample in this context.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Given that the probability that more than $\left( n-4 \right)$ markers in the random sample are found to be non-faulty is less than $0.3$, find the least possible value of $n$.<span class="inline-marks">[2]</span></div></li></ol>
<p>Assume for the remainder of this question that&nbsp;$n=10$.</p>
<p>The samples in $50$ randomly chosen days are collected and the number of faulty markers in each sample is recorded.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the probability that exactly $35$ of the samples each contains at least one faulty marker, given that none of the $50$ samples contain more than $2$ faulty markers each.<span class="inline-marks">[4]</span></div></li></ol>

On average, $7.8\%$ of markers produced by a factory are faulty. Each day, the factory’s quality manager picks a random sample of n markers and inspects the number of faulty markers found in the sample. The number of faulty markers found in the sample is the random variable $X$ and whether the markers are faulty are independent of one another.

Assume for the remainder of this question that $n=10$.

The samples in $50$ randomly chosen days are collected and the number of faulty markers in each sample is recorded.

Minimum Number of Binomial Trials

Population and Random Samples

2022 NJC P2 Q7

<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

Determining Sample Size from Probability Conditions

2022 EJC P2 Q6

<ol class="abc-list"><li><div class="list-item-paragraph">A random variable $X$ has a binomial distribution with $n=10$ and probability of success $p$, where&nbsp;$p&lt;0.5$.</div><ol class="i-list"><li><div class="list-item-paragraph">Given that&nbsp;$\mathrm{P}\left( X=3\,\,\,\mathrm{or}\,\,\,4 \right)=0.2$, write down an equation for the value of $p$, and find this value numerically.<span class="inline-marks">[2]</span></div><div class="list-item-paragraph">It is given that&nbsp;$p=\frac{1}{5}$.</div></li><li><div class="list-item-paragraph">The mean and standard deviation of $X$ are denoted by $\mu $ and $\sigma $ respectively.</div><div class="list-item-paragraph">Find $\mathrm{P}\left( \mu -\sigma &lt; X &lt; \mu +\sigma  \right)$, correct to $2$ decimal places.<span class="inline-marks">[3]</span></div></li></ol></li><li><div class="list-item-paragraph">Mr Chua attempts an online sudoku puzzle each day. The probability that he manages to solve a puzzle on any given day is $0.75$, independently of any other day.</div><ol class="i-list"><li><div class="list-item-paragraph">Find the probability that he solves his third puzzle on the eighth day of his attempt.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the probability that, over a period of $8$ weeks, Mr Chua manages to solve at least $4$ puzzles each week.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

2022 ASRJC P2 Q9

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAzMzgyOTQ4OTEteWRlc2RtbTlvai5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTU2MzU3LCJleHAiOjE3OTM1NTY2NTd9.1YxYkzhSmfkYRaaPaSeHiZpRt3Y_6yZk1K3r4i6edlk" 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.

N2021 P2 Q11

<p>A biscuit factory produces biscuits which are packed into its signature gift boxes. On average, $3\%$ of these gift boxes packed are substandard.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the probability that a randomly chosen batch of $30$ gift boxes will contain at least $2$ gift boxes which are substandard is $0.227$, correct to $3$ significant figures.</div><p style="text-align: right;">[2]</p></li></ol>
<p>The operations unit of the factory decided to pack $30$ gift boxes into one carton before delivery to all its retail outlets.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the probability that out of $10$ randomly chosen cartons, there are more than $4$ cartons with at least $2$ substandard gift boxes.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the probability that out of $10$ randomly chosen cartons, there are at least $2$ substandard gift boxes.</div><p style="text-align: right;">[2]</p></li></ol>
<p>Every week, the owner of each retail outlet will place order with the factory.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Find the maximum number of cartons to be ordered so that the chance of having no substandard gift box is more than $5\%$.</div><p style="text-align: right;">[4]</p></li></ol>

A biscuit factory produces biscuits which are packed into its signature gift boxes. On average, $3\%$ of these gift boxes packed are substandard.

The operations unit of the factory decided to pack $30$ gift boxes into one carton before delivery to all its retail outlets.

Every week, the owner of each retail outlet will place order with the factory.

Maximum Number of Binomial Trials

ASRJC Binomial Distribution Tutorial Q12

<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

<ol class="abc-list"><li data-list-item-id="item-1760254424641-v3pg7gdel" data-list-item-path="item-1760254424641-v3pg7gdel"><ol class="i-list"><li data-list-item-id="subitem-1760254426673-4bp48robu" data-list-item-path="item-1760254424641-v3pg7gdel/subitem-1760254426673-4bp48robu"><div class="list-item-paragraph">It is known that the probability of a customer using e-payment at a hawker stall is $0.25$. A group of customers is chosen at random, find the probability that the ${{7}^{\mathrm{th}}}$ chosen customer is the ${{5}^{\mathrm{th}}}$ customer using e-payment at the hawker stall.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760254427566-3norv1x3a" data-list-item-path="item-1760254424641-v3pg7gdel/subitem-1760254427566-3norv1x3a"><div class="list-item-paragraph">A sample of $40$ customers were randomly chosen from the hawker stall each day. In a month of $30$ days, find the probability that there is at least $15$ days with at most $10$ customers per sample making e-payment.<span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-1760254431759-an5cbjxw9" data-list-item-path="item-1760254431759-an5cbjxw9"><div class="list-item-paragraph">The probability that a hawker uses the online delivery platform Foodgowhere is $p$. A random sample of $~n$ hawkers is taken and the random variable $X$ denotes the number of hawkers in the sample that uses Foodgowhere.</div><ol class="i-list"><li data-list-item-id="subitem-1760254456096-ajuc8x69g" data-list-item-path="item-1760254431759-an5cbjxw9/subitem-1760254456096-ajuc8x69g"><div class="list-item-paragraph">Explain what is meant by a random sample in this context.<span class="inline-marks">[1]</span></div><div class="list-item-paragraph">Assuming that&nbsp;$X$ follows a binomial distribution.</div></li><li data-list-item-id="subitem-1760254456613-hlb5mic08" data-list-item-path="item-1760254431759-an5cbjxw9/subitem-1760254456613-hlb5mic08"><div class="list-item-paragraph">It is given that&nbsp;$\mathrm{P}\left( X\le 1 \right)=\mathrm{0}\mathrm{.05303}$ and the expected number of hawkers using Foodgowhere is $3.96$. </div><div class="list-item-paragraph">Write down two equations satisfied by $p$ and $~n$. Hence find the value of $p$ and $~n$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760254457664-ksry1c6x4" data-list-item-path="item-1760254431759-an5cbjxw9/subitem-1760254457664-ksry1c6x4"><div class="list-item-paragraph">Given that&nbsp;$n=15$, find the set of values of $p$ so that that the most likely number of hawkers in the sample who uses Foodgowehere is $5$.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

2022 ACJC P2 Q11

<p>A factory produces electrical fuses by an automatic process. The fuses are packed in boxes, each box containing $24$ fuses, for dispatch to customers. The production process is under statistical control and it is known from experience that on average $p\%$ of the fuses manufactured are defective. It should be assumed that the number of defective fuses in a box of $24$ fuses follows a binomial distribution.</p>
<ol class="i-list"><li><div class="list-item-paragraph">The probability that a randomly selected box contains at least one defective fuse is $0.518$. Form an equation in $p$ and hence solve for $p$, leaving your answer to $4$ decimal places.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given that $p=3$ and the customer buys $100$ boxes of fuses, find the most likely number of boxes each of which contains less than $1$ defective fuse.</div><p style="text-align: right;">[2]</p></li></ol>

A factory produces electrical fuses by an automatic process. The fuses are packed in boxes, each box containing $24$ fuses, for dispatch to customers. The production process is under statistical control and it is known from experience that on average $p\%$ of the fuses manufactured are defective. It should be assumed that the number of defective fuses in a box of $24$ fuses follows a binomial distribution.

2022 ACJC J2 MYE P1 Q10

<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>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.

Bayes' Theorem

Law of Total Probability

Multistage Binomial Sampling

N2017 P2 Q9

<p>Tomatoes are sold in boxes of $16$. On average, $15\%$ of them are damaged.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, in context, two assumptions needed for the number of damaged tomatoes 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 damaged tomatoes in a box has a binomial distribution.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the most likely number of damaged tomatoes in a randomly chosen box.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the probability that a box contains less than $6$ damaged tomatoes.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">A box contains less than $6$ damaged tomatoes. Find the probability that it contains at least $2$ damaged tomatoes.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Two boxes of tomatoes are randomly selected. Find the probability that one box has at least $2$ damaged tomatoes and the other box has none.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">A market trader buys $80$ boxes of tomatoes. Find the probability that there are more than $60$ boxes with at least $2$ damaged tomatoes, giving your answers to $4$ decimal places.</div><p style="text-align: right;">[3]</p></li></ol>

Tomatoes are sold in boxes of $16$. On average, $15\%$ of them are damaged.

Assume now that the number of damaged tomatoes in a box has a binomial distribution.

2018 PJC BT2 P2 Q8

<p>A manufacturing company produces surgical masks. The surgical masks are randomly packed into boxes of $50$. On average, $15\%$ of the surgical masks are defective. The number of surgical masks that are defective in a box of $50$ pieces is denoted by $X$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, in context, two assumptions needed for the number of defective surgical masks 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 defective surgical masks in a box has a binomial distribution.</p>
<p>In a randomly chosen box of $50$ surgical masks, find the probability that there are</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">more than $8$ defective surgical masks,</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">at most $12$ defective surgical masks given that there are more than $8$ defective surgical masks.</div><p style="text-align: right;">[3]</p></li></ol>
<p>For shipping purposes, the boxes are packed into cartons, with each carton containing $24$ boxes.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Find the probability that, in a randomly chosen carton, there are at least $10$ boxes containing at most $8$ defective surgical masks.</div><p style="text-align: right;">[2]</p></li></ol>
<p>The company also manufactures reusable masks which are packed into packets of $10$. The probability that a reusable mask is defective is $p$. It is known that the modal number of defective reusable masks in a packet is $1$.</p>
<ol class="i-list" start="5"><li><div class="list-item-paragraph">Use this information to find exactly the range of values that $p$ can take.</div><p style="text-align: right;">[3]</p></li></ol>

A manufacturing company produces surgical masks. The surgical masks are randomly packed into boxes of $50$. On average, $15\%$ of the surgical masks are defective. The number of surgical masks that are defective in a box of $50$ pieces is denoted by $X$.

Assume now that the number of defective surgical masks in a box has a binomial distribution.

In a randomly chosen box of $50$ surgical masks, find the probability that there are

For shipping purposes, the boxes are packed into cartons, with each carton containing $24$ boxes.

The company also manufactures reusable masks which are packed into packets of $10$. The probability that a reusable mask is defective is $p$. It is known that the modal number of defective reusable masks in a packet is $1$.

2020 YIJC P2 Q8

<p>A biscuit manufacturer produces both cream and chocolate biscuits. Biscuits are chosen randomly and packed into boxes of $10$. The number of cream biscuits in a box is denoted by $X$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">On average, the proportion of cream biscuits is $p$. Given that $\mathrm{P}\left( X=1\,\,\mathrm{or}\,\,2 \right)=0.15$, write down an equation for the value of $p$. Hence find the value(s) of $p$ numerically.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">It is given instead that the biscuit manufacturer produces $3$ times as many cream biscuits as chocolate biscuits.</div><ol class="i-list"><li><div class="list-item-paragraph">Find the most likely value of $X$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">A random sample of $18$ boxes is taken. Find the probability that at least $3$ but fewer than $7$ boxes have equal numbers of cream and chocolate biscuits.</div><p style="text-align: right;">[3]</p></li></ol></li></ol>
<p>A box of biscuits is sold at $\$10$. The manufacturer gives a discount of $\$2$ per box to its premium customers. The mean and variance of the number of boxes sold per day to each type of customers (assuming independence) are as follows:</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759832233144-khay4yq75mo.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTk4MzIyMzMxNDQta2hheTR5cTc1bW8ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU1NjM1NywiZXhwIjoxNzkzNTU2NjU3fQ.S-JgdQfokm0iu3rMY_TveEHuNtZCYjflK4i9HGYCEqY" width="400" class="aligncenter" /></figure>
<p>Find the approximate probability that the total amount collected per month from the sales of biscuits is not less than $\$255,000$, assuming that there are $30$ days in a month.</p><p style="text-align: right;">[4]</p>

A biscuit manufacturer produces both cream and chocolate biscuits. Biscuits are chosen randomly and packed into boxes of $10$. The number of cream biscuits in a box is denoted by $X$.

A box of biscuits is sold at $\$10$. The manufacturer gives a discount of $\$2$ per box to its premium customers. The mean and variance of the number of boxes sold per day to each type of customers (assuming independence) are as follows:

Find the approximate probability that the total amount collected per month from the sales of biscuits is not less than $\$255,000$, assuming that there are $30$ days in a month.

Linear Combinations of Random Variables

2017 AJC P2 Q8

<p>A bag contains $4$ red, $5$ blue and $6$ green balls. The balls are indistinguishable except for their colour. A trial consists of drawing a ball at random from the bag, noting its colour and replacing it in the bag. A game is played by performing $10$ trials in all.</p>
<p>The player receives $\$0.50$ for every blue ball drawn in a game. He receives nothing if balls of other colours are drawn.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the probability that, in a game, he receives</div><ol class="i-list"><li><div class="list-item-paragraph">exactly $\$3$,</div></li><li><div class="list-item-paragraph">more than $\$2.50$,</div></li></ol></li><li><div class="list-item-paragraph">Find also the amount a player is most likely to receive.</div></li><li><div class="list-item-paragraph">At the start of a tournament, each player plays the above game once. Players who earned more than $\$k$ proceed to the next round. Find the least value of $k$ such that, in a random sample of $10$ players, the probability that all the $10$ players proceed to the next round is less than $0.1$.</div></li></ol>

A bag contains $4$ red, $5$ blue and $6$ green balls. The balls are indistinguishable except for their colour. A trial consists of drawing a ball at random from the bag, noting its colour and replacing it in the bag. A game is played by performing $10$ trials in all.

The player receives $\$0.50$ for every blue ball drawn in a game. He receives nothing if balls of other colours are drawn.

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