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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

<p>In a school with $1200$ students, an average of $1$ out of $20$ students is sick in a week.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">State, in this context, two assumptions needed for the number of students who are sick in a week to be well modelled by a binomial distribution. Explain why one of these assumptions may not be met.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>For the rest of this question, assume that the number of students who are sick in a week has a binomial distribution.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the expected number of students who are sick in a week.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">In a class of $25$ students, find the probability that at least $2$ of them are sick in a week.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">In a sample of $10$ classes with $25$ students each, find the probability that less than five classes have at least $2$ students who are sick in a week, given that there are at least two such classes.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the least value of $n$ such that in a sample of $n$ randomly selected students, the probability that at least $6$ are sick in a week is more than $0.95$.</span><span class="inline-marks">[3]</span></div></li></ol>

In a school with $1200$ students, an average of $1$ out of $20$ students is sick in a week.

State, in this context, two assumptions needed for the number of students who are sick in a week to be well modelled by a binomial distribution. Explain why one of these assumptions may not be met.

For the rest of this question, assume that the number of students who are sick in a week has a binomial distribution.

Find the expected number of students who are sick in a week.

In a class of $25$ students, find the probability that at least $2$ of them are sick in a week.

In a sample of $10$ classes with $25$ students each, find the probability that less than five classes have at least $2$ students who are sick in a week, given that there are at least two such classes.

Find the least value of $n$ such that in a sample of $n$ randomly selected students, the probability that at least $6$ are sick in a week is more than $0.95$.

Minimum Number of Binomial Trials

Binomial Probabilities Using Technology

Binomial Modelling and Applications

Discrete Random Variables (DRV)

<p>One question in a survey is as follows:<br>“Do you use Facebook during office hours at least once a day? Yes or No?''<br>It is known that on average, $7$ out of $9$ staff members answered “yes” to the above question. For the random sample of $60$ staff members selected,</p>
<ol class="i-list"><li data-list-item-id="item-i-2" data-list-item-path="item-i-2"><div class="list-item-paragraph">find the standard deviation of the number of staff members who answer “yes”,<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-ii-2" data-list-item-path="item-ii-2"><div class="list-item-paragraph">find the probability that the number of staff members who answer “yes” exceeds the mean,<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-iii-2" data-list-item-path="item-iii-2"><div class="list-item-paragraph">find the most probable number of staff members who answer “yes”,<span class="inline-marks">[2]</span></div></li></ol>

One question in a survey is as follows:
“Do you use Facebook during office hours at least once a day? Yes or No?''
It is known that on average, $7$ out of $9$ staff members answered “yes” to the above question. For the random sample of $60$ staff members selected,

find the standard deviation of the number of staff members who answer “yes”,

find the probability that the number of staff members who answer “yes” exceeds the mean,

find the most probable number of staff members who answer “yes”,

Mode of a Binomial Distribution

Mean and Variance of a Binomial Distribution

2013 DHS P2 Q10

<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 Models and Assumptions

Determining an Unknown Binomial Success Probability

2013 MJC P2 Q8 [Modified]

<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

Repeated Binomial Experiments

Sample Mean and Its Sampling Distribution

2015 TJC P2 Q9 [Modified]

Sampling

<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>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>Given that $X\sim\mathrm{B}(25,0.35)$, find</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\mathrm{P}(X=10)$</div></li><li><div class="list-item-paragraph">$\mathrm{P}(X&gt;12)$</div></li><li><div class="list-item-paragraph">$\mathrm{P}(8.12 &lt; X\le 14.36)$</div></li><li><div class="list-item-paragraph">$\mathrm{P}(X\ge 17.3)$</div></li><li><div class="list-item-paragraph">$\mathrm{P}\left( \left| X-10 \right|&lt;9 \right)$</div></li><li><div class="list-item-paragraph">$\mathrm{P}(\left| X-12 \right|&gt;3)$</div></li><li><div class="list-item-paragraph">Probability that $X$ lies within one standard deviation from the mean.</div></li><li><div class="list-item-paragraph">Mode of $X$.</div></li></ol>

Given that $X\sim\mathrm{B}(25,0.35)$, find

<p>Given that $X\sim\mathrm{B}(25,0.35)$, find</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\mathrm{P}(X=10)$</div></li><li><div class="list-item-paragraph">$\mathrm{P}(X>12)$</div></li><li><div class="list-item-paragraph">$\mathrm{P}(8.12 < X\le 14.36)$</div></li><li><div class="list-item-paragraph">$\mathrm{P}(X\ge 17.3)$</div></li><li><div class="list-item-paragraph">$\mathrm{P}\left( \left| X-10 \right|<9 \right)$</div></li><li><div class="list-item-paragraph">$\mathrm{P}(\left| X-12 \right|>3)$</div></li><li><div class="list-item-paragraph">Probability that $X$ lies within one standard deviation from the mean.</div></li><li><div class="list-item-paragraph">Mode of $X$.</div></li></ol>

Worked Example 2

<p>A store owner receives a shipment of stationery items, including notebooks, pens, and correction tapes. Historical data indicates that&nbsp;$1\%$ of the notebooks, $2\%$ of the pens, and $4\%$ of the correction tapes are defective. The quality of notebooks, pens and correction tapes is independent of one another. The store owner decides to sell the stationery in $180$ packets, each containing one notebook, two pens and one correction tape. A packet is deemed unsatisfactory if any of the four items is defective.</p>
<ol class="i-list"><li data-list-item-id="item-1761026739916-r8mf46f1p" data-list-item-path="item-1761026739916-r8mf46f1p"><div class="list-item-paragraph">Show that the probability that a randomly selected packet is unsatisfactory is $0.0872$, correct to $3$ significant figures.<span class="inline-marks">[2]</span></div></li></ol>
<p>The number of packets that are unsatisfactory is denoted by $X$. You may assume that&nbsp;$X$ can be modelled by a binomial distribution</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1761026778901-9zeslxr7m" data-list-item-path="item-1761026778901-9zeslxr7m"><div class="list-item-paragraph">Find the probability that there are at least $5$ but less than $10$ packets that are unsatisfactory.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761026779456-k1lne558d" data-list-item-path="item-1761026779456-k1lne558d"><div class="list-item-paragraph">Find the least value of $r$ such that the probability that there are more than $r$ packets that are unsatisfactory is at most $0.12$.<span class="inline-marks">[2]</span></div></li></ol>
<p>Before selling the packets of stationery, he decided to select a sample of $9$ packets to check for unsatisfactory packets.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1761026803727-iw5hgizpy" data-list-item-path="item-1761026803727-iw5hgizpy"><div class="list-item-paragraph">How should the packets be selected? Give a reason for this method of selection.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761026804701-qwibu9mih" data-list-item-path="item-1761026804701-qwibu9mih"><div class="list-item-paragraph">Find the probability that the ninth packet is the third unsatisfactory packet selected.<span class="inline-marks">[2]</span></div></li></ol>

A store owner receives a shipment of stationery items, including notebooks, pens, and correction tapes. Historical data indicates that $1\%$ of the notebooks, $2\%$ of the pens, and $4\%$ of the correction tapes are defective. The quality of notebooks, pens and correction tapes is independent of one another. The store owner decides to sell the stationery in $180$ packets, each containing one notebook, two pens and one correction tape. A packet is deemed unsatisfactory if any of the four items is defective.

The number of packets that are unsatisfactory is denoted by $X$. You may assume that $X$ can be modelled by a binomial distribution

Before selling the packets of stationery, he decided to select a sample of $9$ packets to check for unsatisfactory packets.

Complementary Events

Probabilities of Successive Events

Binomial Probabilities Using the Probability Mass Function

Population and Random Samples

Waiting-Time and Stopping-Rule Probabilities

2024 JPJC P2 Q8

Probability

<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

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 $0.035$.

2025 SAJC P2 Q8

<p><strong>In this question you should state clearly all the distributions that you use, together with the values of the appropriate parameters.</strong></p>
<p>A vegetable seller sells potatoes and tomatoes. The masses of potatoes, $X$ grams, have the distribution $N\left( 175,{{25}^{2}} \right)$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760848429665-vdv25n06e" data-list-item-path="item-1760848429665-vdv25n06e"><div class="list-item-paragraph">Given that&nbsp;$\mathrm{P}\!\left( X \lt 166 \right) = \mathrm{P}\!\left( X \gt m \right)$, find the value of $m$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760848430213-so9erdvf1" data-list-item-path="item-1760848430213-so9erdvf1"><div class="list-item-paragraph">The probability that the mass of a randomly chosen potato differs from the population mean mass by more than $a$ grams is $0.3$. Find the value of $a$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760848430888-jmzdjhech" data-list-item-path="item-1760848430888-jmzdjhech"><div class="list-item-paragraph">Find the expected number of potatoes with mass less than $182$ grams in a randomly chosen batch of $90$ potatoes.<span class="inline-marks">[2]</span></div></li></ol>
<p>The masses of tomatoes, $Y$ grams, have the distribution $N\left( 125,{{15}^{2}} \right)$.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-1760848463273-v7d0z14y0" data-list-item-path="item-1760848463273-v7d0z14y0"><div class="list-item-paragraph">Find the probability that the mass of a randomly chosen tomato is more than $130$ grams.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760848464488-x2hcwqtok" data-list-item-path="item-1760848464488-x2hcwqtok"><div class="list-item-paragraph">A sample of $50$ tomatoes are randomly chosen. Find the probability that exactly $20$ tomatoes in this sample each has a mass more than $130$grams.<span class="inline-marks">[2]</span></div></li></ol>
<p>The selling price of the potatoes is $\$0.32$ per $100$g and the selling price of the tomatoes is $\$0.22$ per $100$g.</p>
<ol class="abc-list" start="6"><li data-list-item-id="item-1760848494908-oint4oz8k" data-list-item-path="item-1760848494908-oint4oz8k"><div class="list-item-paragraph">Find the probability that the price of a randomly chosen potato is greater than the total price of $2$ randomly chosen tomatoes.<span class="inline-marks">[4]</span></div></li></ol>

**In this question you should state clearly all the distributions that you use, together with the values of the appropriate parameters.**

A vegetable seller sells potatoes and tomatoes. The masses of potatoes, $X$ grams, have the distribution $N\left( 175,{{25}^{2}} \right)$.

The masses of tomatoes, $Y$ grams, have the distribution $N\left( 125,{{15}^{2}} \right)$.

The selling price of the potatoes is $\$0.32$ per $100$g and the selling price of the tomatoes is $\$0.22$ per $100$g.

Normal Probabilities and Quantiles

Symmetry and Properties of the Normal Distribution

Linear Combinations of Independent Normal Variables

2024 YIJC P2 Q12

Continuous Random Variables

<p>In a certain airline route, there are a maximum of $250$ seats and each ticket is sold at $\$350$. If costs $\$1050$ for the airline for every overbooked passenger if they show up for the flight for compensations. Assume that a booked passenger will not show for the flight with probability $8\%$ and that the number of passengers who show up for each flight follows a binomial distribution. The airline has sold $254$ tickets for a particular flight.</p>
<ol class="abc-list"><li data-list-item-id="item-1760703015096-6nric3mxc" data-list-item-path="item-1760703015096-6nric3mxc"><div class="list-item-paragraph">What is the probability that there will not be enough seats for the flight?</div></li><li data-list-item-id="item-1760703016026-3n3fp80t6" data-list-item-path="item-1760703016026-3n3fp80t6"><div class="list-item-paragraph">Find the expected amount of revenue for selling $254$ tickets.</div></li></ol>
<p>Assume that the airline sells $n$ tickets with $n&gt;250$ as they decide to overbook.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-1760703030784-3c7yyg38d" data-list-item-path="item-1760703030784-3c7yyg38d"><div class="list-item-paragraph"><p>Find the expected amount of revenue for selling $n$ tickets in the form of $\mathrm{f}\left( n \right)\mathrm{P}\left( X\le 250 \right)+\sum\limits_{r=251}^{n}{\left( \left[ \mathrm{f}\left( n \right)-\mathrm{g}\left( r \right) \right]P\left( X=r \right) \right)}$, where&nbsp;$\mathrm{f}\left( n \right)$ and $\mathrm{g}\left( r \right)$ are functions in terms of $n$ and $r$ respectively to be determine.</p></div></li><li data-list-item-id="item-1760703031476-r55hw6fh6" data-list-item-path="item-1760703031476-r55hw6fh6"><div class="list-item-paragraph">Hence, find the number of tickets to be overbooked in order to maximise the revenue.</div></li></ol>

In a certain airline route, there are a maximum of $250$ seats and each ticket is sold at $\$350$. If costs $\$1050$ for the airline for every overbooked passenger if they show up for the flight for compensations. Assume that a booked passenger will not show for the flight with probability $8\%$ and that the number of passengers who show up for each flight follows a binomial distribution. The airline has sold $254$ tickets for a particular flight.

Assume that the airline sells $n$ tickets with $n>250$ as they decide to overbook.

Expectation and Variance of Discrete Random Variables

Expected Payoffs and Fair Games

Prelims Revision Set: Problems involving Overbooking Q2

<p>Airlines routinely overbook flights based on the expectation that some fraction of booked passengers will not show up for each flight.For a particular flight that has a maximum capacity of $300$ seats, the airline has sold $312$ tickets.Assume that a booked passenger will not show for the flight with probability $5\%$ and that the number of passengers who show up for each flight follows a binomial distribution.</p>
<ol class="i-list"><li data-list-item-id="item-1760702896205-j2u1p6b9e" data-list-item-path="item-1760702896205-j2u1p6b9e"><div class="list-item-paragraph">What is the probability that there will not be enough seats for the flight?</div></li><li data-list-item-id="item-1760702896963-chmaz9d5l" data-list-item-path="item-1760702896963-chmaz9d5l"><div class="list-item-paragraph">Find the probability that the flight has empty seat(s).</div></li><li data-list-item-id="item-1760702898022-vyceql7w9" data-list-item-path="item-1760702898022-vyceql7w9"><div class="list-item-paragraph">Find the number of tickets that can be sold to ensure that the probability of overbooking is less than $3\%$.</div></li></ol>

Airlines routinely overbook flights based on the expectation that some fraction of booked passengers will not show up for each flight.For a particular flight that has a maximum capacity of $300$ seats, the airline has sold $312$ tickets.Assume that a booked passenger will not show for the flight with probability $5\%$ and that the number of passengers who show up for each flight follows a binomial distribution.

Maximum Number of Binomial Trials

Prelims Revision Set: Problems involving Overbooking Q1

<p>The results of an experiment show that the discrete random variable $X$ has a binomial distribution with mean $1.2$ and variance $1.18$.</p>
<p>Find the smallest integer $r$ such that $\mathrm{P}(X &lt; r)&gt;0.85$.</p>

The results of an experiment show that the discrete random variable $X$ has a binomial distribution with mean $1.2$ and variance $1.18$.

Find the smallest integer $r$ such that $\mathrm{P}(X < r)>0.85$.

<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 $3$.**

2020 MI P2 Q9

<p>A nursery supplies plants to Evergreen Pte. Ltd. The heights, in centimetres, of plants supplied may be taken to follow the distribution$\mathrm{N}\left( \mu ,{{\sigma }^{2}} \right)$.</p>
<p>Over the years, Evergreen rejects $6\%$ of the plants supplied as too short and $4\%$ as too tall. It is known that Evergreen only accepts plants of heights between $25$ cm and $91$ cm.</p>
<p>Show that $\sigma =20$ correct to the nearest integer. Hence find the median height of the plants accepted by Evergreen.</p><p style="text-align: right;">[5]</p>
<p>Evergreen inspects the plants supplied in batches of $10$. Find the probability that a total of at most $4$ plants are rejected in two randomly chosen batches.</p><p style="text-align: right;">[2]</p>
<p>Three children, Anna, Bella and Cinderella, each chooses a plant at random from the nursery. Find the probability that the average height of the three plants is at least twice the average height of the plants chosen by Anna and Bella.</p><p style="text-align: right;">[3]</p>

A nursery supplies plants to Evergreen Pte. Ltd. The heights, in centimetres, of plants supplied may be taken to follow the distribution$\mathrm{N}\left( \mu ,{{\sigma }^{2}} \right)$.

Over the years, Evergreen rejects $6\%$ of the plants supplied as too short and $4\%$ as too tall. It is known that Evergreen only accepts plants of heights between $25$ cm and $91$ cm.

Show that $\sigma =20$ correct to the nearest integer. Hence find the median height of the plants accepted by Evergreen.

Evergreen inspects the plants supplied in batches of $10$. Find the probability that a total of at most $4$ plants are rejected in two randomly chosen batches.

Three children, Anna, Bella and Cinderella, each chooses a plant at random from the nursery. Find the probability that the average height of the three plants is at least twice the average height of the plants chosen by Anna and Bella.

Normal Models for Sample Means and Averages

Finding Unknown Normal Mean and Standard Deviation

2015 AJC P2 Q7

<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>
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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.

Probability Using Permutations and Combinations

2023 DHS P2 Q8

<p>A dart board is designed with an inner radius of $r$ cm and an outer radius of $15$ cm, as shown in the diagram below.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760448997302-lpx6r8fk7mi.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA0NDg5OTczMDItbHB4NnI4Zms3bWkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0ODAxNiwiZXhwIjoxNzkzNTQ4MzE2fQ.34UsNgVP89Xvv4PKFycNBiikCijN858F6ZeYiiPWXl0" style="max-width: min(370px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>A dart throw is considered a ‘hit’ if it lands within the inner circle. It is assumed that any dart throw lands randomly on the dart board.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that the probability of a ‘hit’ is $\frac{{{r}^{2}}}{225}$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Wei plays a round of $8$ dart throws.</div><ol class="i-list"><li><div class="list-item-paragraph">The probability that Wei obtains more than six ‘hits’ is at most $0.08$. Write down an inequality satisfied by $r$. Hence find the range of possible values of $r$.<span class="inline-marks">[2]</span></div><div class="list-item-paragraph">It is now given that&nbsp;$r=6$</div></li><li><div class="list-item-paragraph">Find the probability that Wei’s fifth throw is the second time that he obtains a ‘hit’.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the least number of throws that Wei needs to attempt such that the probability of obtaining at least $1$ ‘hit’ from the attempts is at least $0.7$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Chan also plays a round of $8$ dart throws. The number of ‘hits’ Chan obtained is more than the expected number of ‘hits’ Wei obtained. Find the probability that the number of ‘hits’ Chan obtained is fewer than twice the expected number of ‘hits’ Wei obtained.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

A dart board is designed with an inner radius of $r$ cm and an outer radius of $15$ cm, as shown in the diagram below.

A dart throw is considered a ‘hit’ if it lands within the inner circle. It is assumed that any dart throw lands randomly on the dart board.

2023 YIJC P2 Q9

<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.

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