A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>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.

Binomial Probabilities Using Technology

Probabilities Involving Multiple Binomial Variables

Repeated Binomial Experiments

Discrete Random Variables (DRV)

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

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

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

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

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

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

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

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

Conditional Probability

Normal Probabilities and Quantiles

Assumptions for Sampling

Linear Transformations of a Normal Variable

Normal Models for Sample Means and Averages

Linear Combinations of Independent Normal Variables

Assessing the Suitability of a Normal Model

Sample Mean and Its Sampling Distribution

2025 HCI P2 Q10

Probability

Continuous Random Variables

Sampling

<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 $0.035$.

Binomial Models and Assumptions

Probabilities of Successive Events

Binomial Probabilities Using the Probability Mass Function

Determining an Unknown Binomial Success Probability

Waiting-Time and Stopping-Rule Probabilities

2025 SAJC P2 Q8

<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.

Maximum Number of Binomial Trials

2018 SRJC P2 Q12

<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.

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

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>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.

Mode of a Binomial Distribution

2018 PJC BT2 P2 Q8

<p>At a supermart, the option of using cashless payment for purchases is available to customers. On average, $p\%$ of the customers use cashless payment.</p>
<p>The manager of the supermart selects a sample of $30$ customers for a survey.</p>
<ol class="i-list"><li data-list-item-id="item-1759654852295-snio4izf3" data-list-item-path="item-1759654852295-snio4izf3"><div class="list-item-paragraph">The probability of not more than $1$ of customers surveyed use cashless payment for their purchases is $\mathrm{0}\mathrm{.245}$. Write down an equation in terms of $p$. Find $p$, giving your answer correct to $3$ decimal places.<span class="inline-marks">[3]</span></div></li></ol>
<p>For the remaining parts of this question, use $p=30$.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1759654868015-79amydrsq" data-list-item-path="item-1759654868015-79amydrsq"><div class="list-item-paragraph">Find the probability that the last customer of the $30$ selected for survey is the ${{10}^{\mathrm{th}}}$ customer who uses a cashless payment for his purchase.<span class="inline-marks">[2]</span></div></li></ol>
<p>Another manager of the supermart also selects a sample of $30$ customers.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1759654891857-f926hr57n" data-list-item-path="item-1759654891857-f926hr57n"><div class="list-item-paragraph">Given that there are a total of less than $16$ customers using cashless payment for their purchases from both samples, find the probability that one of the sample has more than $12$ customers using cashless payment for their purchases.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759654892728-gf0mqobo0" data-list-item-path="item-1759654892728-gf0mqobo0"><div class="list-item-paragraph"><p>State one assumption you have made in the calculation in part <strong>(iii)</strong>.</p><span class="inline-marks">[1]</span></div></li></ol>

At a supermart, the option of using cashless payment for purchases is available to customers. On average, $p\%$ of the customers use cashless payment.

The manager of the supermart selects a sample of $30$ customers for a survey.

For the remaining parts of this question, use $p=30$.

Another manager of the supermart also selects a sample of $30$ customers.

2018 RVHS P2 Q7

<p>A company produces drinking mugs. It is known that, on average, $8\%$ of the mugs are faulty. Each day the quality manager collects $50$ of the mugs at random and checks them; the number of faulty mugs found is the random variable $F$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, in the context of the question, two assumptions needed to model $F$ by a binomial distribution.</div><p style="text-align: right;">[2]</p></li></ol>
<p>You are now given that $F$ can be modelled by a binomial distribution.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the probability that, on a randomly chosen day, at least $7$ faulty mugs are found.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">The number of faulty mugs produced each day is independent of other days. Find the probability that, in a randomly chosen working week of $5$ days, at least $7$ faulty mugs are found on no more than $2$ days.</div><p style="text-align: right;">[2]</p></li></ol>
<p>The company also makes saucers. The number of faulty saucers also follows a binomial distribution. The probability that a saucer is faulty is $p$. Faults on saucers are independent of faults on mugs.</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 $10$ saucers, exactly $2$ are faulty.</div><p style="text-align: right;">[1]</p></li></ol>
<p>The mugs and saucers are sold in sets of $2$ randomly chosen mugs and $2$ randomly chosen saucers. The probability that a set contains at most $1$ faulty item is $0.97$.</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 produces drinking mugs. It is known that, on average, $8\%$ of the mugs are faulty. Each day the quality manager collects $50$ of the mugs at random and checks them; the number of faulty mugs found is the random variable $F$.

You are now given that $F$ can be modelled by a binomial distribution.

The company also makes saucers. The number of faulty saucers also follows a binomial distribution. The probability that a saucer is faulty is $p$. Faults on saucers are independent of faults on mugs.

The mugs and saucers are sold in sets of $2$ randomly chosen mugs and $2$ randomly chosen saucers. The probability that a set contains at most $1$ faulty item is $0.97$.

N2019 P2 Q7

<p>A manufacturing company produces semi-conductor chips. On average, for every $1000$ semi-conductor chips that were manufactured, $p$ would be faulty. The semi-conductor chips are packaged in boxes of $50$ before selling to end users.</p>
<ol class="i-list"><li data-list-item-id="item-1759647323162-fksihz8ku" data-list-item-path="item-1759647323162-fksihz8ku"><div class="list-item-paragraph">State, in context two assumptions needed for the number of faulty semi-conductor chips to be well modelled by binomial distribution.<span class="inline-marks">[2]</span></div></li></ol>
<p>Assume now the assumptions in <strong>(i)</strong> do in fact hold.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1759647337795-1k5q2lxqu" data-list-item-path="item-1759647337795-1k5q2lxqu"><div class="list-item-paragraph">It is given that in a box of $50$ semi-conductor chips, the probability that it contains at most $1$ faulty semi-conductor chip is $0.336$. Write down an equation for&nbsp;$p$ and solve for this value numerically, correct to the nearest integer.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1759647338484-ehc1923rq" data-list-item-path="item-1759647338484-ehc1923rq"><div class="list-item-paragraph">Using the value of $p$ found in <strong>(ii)</strong>, find the probability that among the two randomly selected boxes of semi-conductor chips, one box has at most $3$ faulty semi-conductor chips and the other box has exactly $5$ faulty semi-conductor chips.<span class="inline-marks">[2]</span></div></li></ol>

A manufacturing company produces semi-conductor chips. On average, for every $1000$ semi-conductor chips that were manufactured, $p$ would be faulty. The semi-conductor chips are packaged in boxes of $50$ before selling to end users.

Assume now the assumptions in **(i)** do in fact hold.

2020 SAJC P2 Q7

<p>Emergency flares are simple signalling devices similar to fireworks and they are designed to communicate a much more direct message in an emergency, for example, distress at sea.</p>
<p>A company categorised their stocks of emergency flares as $1$-year old, $5$-year old and $10$-year old.</p>
<p>The probabilities of successful firing of $1$-year old, $5$-year old and $10$-year old emergency flares are $0.995$, $0.970$ and $0.750$ respectively.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that, out of $100$ randomly chosen $1$-year old flares, at most $2$ fail to fire successfully.</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">One-year old flares are packed into boxes of $100$ flares. Find the probability that, out of $50$ randomly chosen boxes of $1$- year old flares, not more than $48$ of these boxes will have at most $2$ flares that will fail to fire successfully in each box.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Seven flares are chosen at random, of which one is $5$ years old and six are $10$ years old. Find the probability that</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the $5$-year old flare fails to fire successfully and at least $4$ of the $10$-year old flares fire successfully,</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">at least $4$ of the $7$ flares fire successfully.</span></div><p style="text-align: right;">[3]</p></li></ol></li></ol>

Emergency flares are simple signalling devices similar to fireworks and they are designed to communicate a much more direct message in an emergency, for example, distress at sea.

A company categorised their stocks of emergency flares as $1$-year old, $5$-year old and $10$-year old.

The probabilities of successful firing of $1$-year old, $5$-year old and $10$-year old emergency flares are $0.995$, $0.970$ and $0.750$ respectively.

Find the probability that, out of $100$ randomly chosen $1$-year old flares, at most $2$ fail to fire successfully.

One-year old flares are packed into boxes of $100$ flares. Find the probability that, out of $50$ randomly chosen boxes of $1$- year old flares, not more than $48$ of these boxes will have at most $2$ flares that will fail to fire successfully in each box.

Seven flares are chosen at random, of which one is $5$ years old and six are $10$ years old. Find the probability that

2019 NYJC P2 Q6

<p>In this question you should state the parameters of any normal distributions you use.</p>
<p>In a high school with a large population, the test scores of the boys, $B$, are normally distributed about a mean of $55$ marks with standard deviation $~11$ marks and the test scores of the girls, $G$, are normally distributed about a mean of $58$ marks with standard deviation $8$ marks.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">A boy and a girl are chosen at random. Find the probability that the difference between their scores is at least $3$ marks.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">The probability that the mean score of four randomly chosen boys and five randomly chosen girls is less than $m$ is more than $0.85497$. Find the range of values of $m$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">A student whose test score exceeds $70$ marks is classified as a smart student. Three boys and two girls are chosen at random. Find the probability that there are exactly two smart students among them.</span></div><p style="text-align: right;">[4]</p></li></ol>

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

In a high school with a large population, the test scores of the boys, $B$, are normally distributed about a mean of $55$ marks with standard deviation $~11$ marks and the test scores of the girls, $G$, are normally distributed about a mean of $58$ marks with standard deviation $8$ marks.

A boy and a girl are chosen at random. Find the probability that the difference between their scores is at least $3$ marks.

The probability that the mean score of four randomly chosen boys and five randomly chosen girls is less than $m$ is more than $0.85497$. Find the range of values of $m$.

A student whose test score exceeds $70$ marks is classified as a smart student. Three boys and two girls are chosen at random. Find the probability that there are exactly two smart students among them.

Solving Normal Probability Inequalities

2022 DHS J2 MYE P2 Q4

<p>A factory produces a particular type of electronic component. The probability of a component being acceptable is $0.96$. The components are packed in boxes of $24$. A box is rejected if it contains fewer than $22$ acceptable components.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that the probability that a randomly chosen box is rejected is $0.0693$.</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">$5$ boxes are chosen at random. Find the probability that two of the boxes each contain exactly $22$ acceptable components and one box is rejected.</span></div><p style="text-align: right;">[3]</p></li></ol>
<p>The factory produces $70$ boxes each day. The production is being monitored for $n$ days, where $n$ is large.</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the expected number of rejected boxes in a day.</span></div><p style="text-align: right;">[2]</p></li></ol>

A factory produces a particular type of electronic component. The probability of a component being acceptable is $0.96$. The components are packed in boxes of $24$. A box is rejected if it contains fewer than $22$ acceptable components.

Show that the probability that a randomly chosen box is rejected is $0.0693$.

$5$ boxes are chosen at random. Find the probability that two of the boxes each contain exactly $22$ acceptable components and one box is rejected.

The factory produces $70$ boxes each day. The production is being monitored for $n$ days, where $n$ is large.

Find the expected number of rejected boxes in a day.

Probability Using Permutations and Combinations

Mean and Variance of a Binomial Distribution

2013 TJC P2 Q12

<p>In order to be offered a scholarship, a candidate has to pass two rounds of interview. (It is assumed that all interviewers’ decisions are independent.)</p>
<p>In the first round, there will be a panel of $10$ interviewers and the probability of each interviewer passing a candidate is $0.9$. The candidate fails to qualify for the second round if more than one interviewer decides not to pass him or her.</p>
<p>Find the probability that a candidate passes the first round of interview.</p>
<p>In the second round, there will be a panel of $5$ interviewers and the probability of each interviewer passing a candidate is $0.8$. The candidate is offered a scholarship only if all interviewers pass him or her in the second round.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that the probability that the candidate is offered the scholarship is $0.241$, correct to three significant figures.</span></div></li><li><div class="list-item-content"><span class="list-item-text">There are $n$ candidates going for the interviews. Find the smallest $n$ such that there is at least a $98\%$ chance of $2$ or more candidates being offered the scholarships.</span></div></li></ol>

In order to be offered a scholarship, a candidate has to pass two rounds of interview. (It is assumed that all interviewers’ decisions are independent.)

In the first round, there will be a panel of $10$ interviewers and the probability of each interviewer passing a candidate is $0.9$. The candidate fails to qualify for the second round if more than one interviewer decides not to pass him or her.

Find the probability that a candidate passes the first round of interview.

In the second round, there will be a panel of $5$ interviewers and the probability of each interviewer passing a candidate is $0.8$. The candidate is offered a scholarship only if all interviewers pass him or her in the second round.

Show that the probability that the candidate is offered the scholarship is $0.241$, correct to three significant figures.

There are $n$ candidates going for the interviews. Find the smallest $n$ such that there is at least a $98\%$ chance of $2$ or more candidates being offered the scholarships.

Minimum Number of Binomial Trials

2010 AJC P2 Q9

<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="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that, in a game, he receives more than $\$2.50$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">If $20$ players are randomly chosen, find the expected number of players who will receive more than $\$2.50$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Find also the amount a player is most likely to receive.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Tom and Jerry both played the game. Given that they both received more than $\$2.50$, find the probability that Tom received at least $\$1.50$ more than Jerry.</span></div><p style="text-align: right;">[3]</p></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.

Find the probability that, in a game, he receives more than $\$2.50$.

If $20$ players are randomly chosen, find the expected number of players who will receive more than $\$2.50$.

Find also the amount a player is most likely to receive.

Tom and Jerry both played the game. Given that they both received more than $\$2.50$, find the probability that Tom received at least $\$1.50$ more than Jerry.

Math Question Bank