A Math

A Math (IGCSE)

E Math

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

<p>Given that&nbsp;$X \sim \mathrm{N}(123, 45^2)$,</p>
<p>find the range of values of $k$ such that</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(X \le k) &gt; 0.28$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(X &gt; k) &gt; 0.777$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(|X - 123| &gt; k) \le 0.444$</span></div></li></ol>

Given that $X \sim \mathrm{N}(123, 45^2)$,

find the range of values of $k$ such that

Normal Distribution

<p>The random variable $X$ has the distribution $ \mathrm{N}(50, 8^2)$. Given that&nbsp;$X_1$ and $X_2$ are two independent observations of $X$, find</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$P(X_1 + X_2 &gt; 120)$,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">$P(X_1 &gt; X_2 + 15)$.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>The random variable $Y$ is related to $X$ by the formula $Y = aX + b$, where $a$ and $b$ are constants with $a &gt; 0$. Given that $ \mathrm{P}(Y &lt; 74) = \mathrm{P}(Y &gt; 146) = 0.0668$, find the values of $ \mathrm{E}(Y)$ and $ \mathrm{Var}(Y)$, and hence find the values of $a$ and $b$.<span class="inline-marks">[7]</span></p>

The random variable $X$ has the distribution $ \mathrm{N}(50, 8^2)$. Given that $X_1$ and $X_2$ are two independent observations of $X$, find

The random variable $Y$ is related to $X$ by the formula $Y = aX + b$, where $a$ and $b$ are constants with $a > 0$. Given that $ \mathrm{P}(Y < 74) = \mathrm{P}(Y > 146) = 0.0668$, find the values of $ \mathrm{E}(Y)$ and $ \mathrm{Var}(Y)$, and hence find the values of $a$ and $b$.

RI Normal Distribution Tutorial Q4

<p>The weights of a randomly chosen $10$-month old male baby, $X$ kg, and a randomly chosen female baby, $Y$ kg, are modelled as having independent normal distributions with means and standard deviations as shown in the table.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 450px;"><colgroup><col style="width: 150px;"><col style="width: 150px;"><col style="width: 150px;"></colgroup><thead><tr><th style="text-align: left; --cell-justify: flex-start"></th><th style="text-align: center; --cell-justify: center">Mean</th><th style="text-align: center; --cell-justify: center">Standard Deviation</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$X$</td><td style="text-align: center; --cell-justify: center">$9.2$</td><td style="text-align: center; --cell-justify: center">$0.8$</td></tr><tr><td style="text-align: center; --cell-justify: center">$Y$</td><td style="text-align: center; --cell-justify: center">$8.5$</td><td style="text-align: center; --cell-justify: center">$0.5$</td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that the difference between thrice the weight of a male baby and the total weight of five female babies does not exceed $10$ kg.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that the mean weight of four randomly chosen male babies is at least $0.5$ kg more than the mean weight of three randomly chosen female babies.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">The probability that the total weight of two males and one female babies is less than $m$ kg is at least $0.9$. Find the least integral value of $m$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Four female babies are selected at random. Calculate the probability that exactly two of them weigh more than $9$ kg and exactly one of them weighs less than $8$ kg.</span><span class="inline-marks">[3]</span></div></li></ol>

The weights of a randomly chosen $10$-month old male baby, $X$ kg, and a randomly chosen female baby, $Y$ kg, are modelled as having independent normal distributions with means and standard deviations as shown in the table.

Find the probability that the difference between thrice the weight of a male baby and the total weight of five female babies does not exceed $10$ kg.

Find the probability that the mean weight of four randomly chosen male babies is at least $0.5$ kg more than the mean weight of three randomly chosen female babies.

The probability that the total weight of two males and one female babies is less than $m$ kg is at least $0.9$. Find the least integral value of $m$.

Four female babies are selected at random. Calculate the probability that exactly two of them weigh more than $9$ kg and exactly one of them weighs less than $8$ kg.

<p>A market stall sells two types of fruit, watermelons and mangos. The masses, in kg, of each type of fruit have independent normal distributions. The means and standard deviations of these distributions, and the selling prices, in $\$ $ per kg, are shown in the following table.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 560px;"><colgroup><col style="width: 140px;"><col style="width: 140px;"><col style="width: 140px;"><col style="width: 140px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Mean (kg)</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Standard deviation (kg)</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Price ($\$ $ per kg)</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Watermelon</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$4.2$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$0.6$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$4$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Mango</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$0.8$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$0.1$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$2.50$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that two randomly chosen watermelons each have a mass between $3$ and $5$ kg.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">$20\%$ of watermelons weigh less than $c$ kg. Find $c$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that the total mass of $2$ randomly chosen watermelons is within $\pm 0.5$ kg of the total mass of $10$ randomly chosen mangoes.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that the total cost of $2$ randomly chosen watermelons and $3$ randomly chosen mangoes is more than $\$ 42$.</span><span class="inline-marks">[4]</span></div></li></ol>

A market stall sells two types of fruit, watermelons and mangos. The masses, in kg, of each type of fruit have independent normal distributions. The means and standard deviations of these distributions, and the selling prices, in $\$ $ per kg, are shown in the following table.

Find the probability that two randomly chosen watermelons each have a mass between $3$ and $5$ kg.

Find the probability that the total mass of $2$ randomly chosen watermelons is within $\pm 0.5$ kg of the total mass of $10$ randomly chosen mangoes.

Find the probability that the total cost of $2$ randomly chosen watermelons and $3$ randomly chosen mangoes is more than $\$ 42$.

2024 YIJC H1 Q12

<p>Mr Hsu takes the train to office for work every weekday and is supposed to arrive at office by $7.30$ a.m. On average, he reaches the train station at $6.00$ a.m. His arrival time at the train station is normally distributed with a standard deviation of $10$ minutes. Every morning, there are only two trains, each departing $5$ minutes apart, with the first train departing at $6.10$ a.m. sharp. The time taken for the train journey follows a normal distribution with mean $60$ minutes and standard deviation $4$ minutes.</p>
<p>After alighting from the train, the time taken to walk from the train station to his office follows a normal distribution with mean $10$ minutes and standard deviation $3$ minutes. Mr Hsu will be late for work if he misses the second train or arrives at office after $7.30$ a.m. Assume that all travelling and waiting times are independent of each other.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">On a randomly chosen day, find the probability that Mr Hsu takes the first train and is late for work.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">On a randomly chosen day, show that the probability that Mr Hsu is late for work is $0.101$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">On a randomly chosen day, Mr Hsu has a briefing to attend before work. Given that he takes the first train, find the earliest starting time of the briefing (correct to the nearest minutes) for which the probability that he is late for the briefing is not more than $0.1$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">In a randomly chosen year of $260$ work days, find the expected number of days that Mr Hsu arrive late for work.</span><span class="inline-marks">[2]</span></div></li></ol>

Mr Hsu takes the train to office for work every weekday and is supposed to arrive at office by $7.30$ a.m. On average, he reaches the train station at $6.00$ a.m. His arrival time at the train station is normally distributed with a standard deviation of $10$ minutes. Every morning, there are only two trains, each departing $5$ minutes apart, with the first train departing at $6.10$ a.m. sharp. The time taken for the train journey follows a normal distribution with mean $60$ minutes and standard deviation $4$ minutes.

After alighting from the train, the time taken to walk from the train station to his office follows a normal distribution with mean $10$ minutes and standard deviation $3$ minutes. Mr Hsu will be late for work if he misses the second train or arrives at office after $7.30$ a.m. Assume that all travelling and waiting times are independent of each other.

On a randomly chosen day, find the probability that Mr Hsu takes the first train and is late for work.

On a randomly chosen day, show that the probability that Mr Hsu is late for work is $0.101$.

On a randomly chosen day, Mr Hsu has a briefing to attend before work. Given that he takes the first train, find the earliest starting time of the briefing (correct to the nearest minutes) for which the probability that he is late for the briefing is not more than $0.1$.

In a randomly chosen year of $260$ work days, find the expected number of days that Mr Hsu arrive late for work.

2024 DHS H1 Q11

<p>The random variable $X$ has distribution $N(\mu, \sigma^2)$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Given that&nbsp;$P(X &gt; k) = 0.45$, find the value of $P(X &gt; 2\mu - k)$.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is given that&nbsp;$\mu = 4$ and $P(1 &lt; X &lt; 4) = 0.2475$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the value of $\sigma$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Draw a sketch to show the distribution of $X$ for&nbsp;$x$ between $-11$ and $19$.<span class="inline-marks">[2]</span></div></li></ol>

The random variable $X$ has distribution $N(\mu, \sigma^2)$.

Given that $P(X > k) = 0.45$, find the value of $P(X > 2\mu - k)$.

It is given that $\mu = 4$ and $P(1 < X < 4) = 0.2475$.

Draw a sketch to show the distribution of $X$ for $x$ between $-11$ and $19$.

2025 VJC P2 Q9

<p>In this question you should state the parameters of any normal distributions you use.<br><br>A machine grades apples according to their mass. Apples with mass exceeding $120$ g are labelled as "large" and apples with mass less than $80$ g are labelled as "small". A large batch of apples is graded and it is found that&nbsp;$15\%$ are "large" and $10\%$ are "small". It is known that the masses of the apples are normally distributed.</p>
<ol class="abc-list"><li data-list-item-id="item-7a" data-list-item-path="item-7a"><div class="list-item-paragraph">Find the mean mass of a randomly chosen apple from the batch and show that the standard deviation is $17.3$ g, correct to $3$ significance figures.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-7b" data-list-item-path="item-7b"><div class="list-item-paragraph">Two apples are chosen in random. Find the probability that the difference in their masses is at least $20$ g.<span class="inline-marks">[3]</span></div></li></ol>
<p>The apples are labelled and packed into boxes. Each box contains $20$ apples.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-7c" data-list-item-path="item-7c"><div class="list-item-paragraph">Apples with mass between $110$ g and $140$ g are labelled as "premium". Find the expected number of "premium" apples in a randomly chosen box.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-7d" data-list-item-path="item-7d"><div class="list-item-paragraph">Find the probability that the mean mass of apples in a randomly chosen box is at most $105$ g.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-7e" data-list-item-path="item-7e"><div class="list-item-paragraph">The boxes of apples are sold by weight at a price of $\$2.90$ per kilogram. Find the probability that the total price of $3$ randomly chosen boxes of apples exceeds $\$18$.<span class="inline-marks">[3]</span></div></li></ol>

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

A machine grades apples according to their mass. Apples with mass exceeding $120$ g are labelled as "large" and apples with mass less than $80$ g are labelled as "small". A large batch of apples is graded and it is found that $15\%$ are "large" and $10\%$ are "small". It is known that the masses of the apples are normally distributed.

Find the mean mass of a randomly chosen apple from the batch and show that the standard deviation is $17.3$ g, correct to $3$ significance figures.

Two apples are chosen in random. Find the probability that the difference in their masses is at least $20$ g.

The apples are labelled and packed into boxes. Each box contains $20$ apples.

Apples with mass between $110$ g and $140$ g are labelled as "premium". Find the expected number of "premium" apples in a randomly chosen box.

Find the probability that the mean mass of apples in a randomly chosen box is at most $105$ g.

The boxes of apples are sold by weight at a price of $\$2.90$ per kilogram. Find the probability that the total price of $3$ randomly chosen boxes of apples exceeds $\$18$.

2025 NYJC P2 Q11

<p>The waiting times for customers to get their pastries from a popular bakery is modelled by a normal distribution with mean $\mu$ minutes and standard deviation $\sigma$ minutes. On average, $10\%$ of customers wait for more than $25$ minutes and the same percentage of customers wait for less than $5$ minutes.</p>
<ol class="i-list"><li data-list-item-id="item-q2-i" data-list-item-path="item-q2-i"><div class="list-item-paragraph">Find the values of $\mu$ and $\sigma$, leaving your answers to the nearest minute.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-q2-ii" data-list-item-path="item-q2-ii"><div class="list-item-paragraph">Explain if the normal distribution is a suitable model for the waiting times for customers to get pastries from this bakery.<span class="inline-marks">[1]</span></div></li></ol>

The waiting times for customers to get their pastries from a popular bakery is modelled by a normal distribution with mean $\mu$ minutes and standard deviation $\sigma$ minutes. On average, $10\%$ of customers wait for more than $25$ minutes and the same percentage of customers wait for less than $5$ minutes.

Find the values of $\mu$ and $\sigma$, leaving your answers to the nearest minute.

Explain if the normal distribution is a suitable model for the waiting times for customers to get pastries from this bakery.

2025 MI P2 Q6

<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 company produces two types of yarn balls with two machines. Machine A produces balls of 2-ply yarn. Over a period of time, it is found that&nbsp;$80\%$ of the balls of yarn from Machine A have lengths less than $340$ metres and $10\%$ of the balls of yarn from Machine A have lengths less than $280$ metres.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Assuming that the lengths, in metres, of yarn in balls of 2-ply yarn from Machine A follows the distribution $N(\mu, \sigma^2)$, find $\mu$ and show that&nbsp;$\sigma = 28.3$.<span class="inline-marks">[3]</span></div></li></ol>
<p>For the rest of the question let&nbsp;$\mu = 300$ and $\sigma = 20$.</p>
<p>Machine B produces balls of 3-ply yarn. The lengths, in metres, of yarn in balls of 3-ply yarn from Machine B follows the distribution $N(220, 6^2)$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Five balls of 3-ply yarn from Machine B are randomly chosen. Find the probability that two of them have lengths more than 230 m and three of them have lengths less than 215 m.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">The probability that two randomly chosen balls of 2-ply yarn from Machine A and three randomly chosen balls of 3-ply yarn from Machine B have total length exceeding $k$ metres is 0.385. Find the value of $k$, correct to 2 decimal places.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph"><p>A technician changed the settings on the two machines. The lengths of yarn in balls of 2-ply yarn from Machine A is reduced by $9\%$ and the lengths of yarn in balls of 3-ply yarn from Machine B is reduced by $5\%$. Find the probability that the total length of yarn in three randomly chosen balls of 2-ply yarn from Machine A is within 50 metres of the total length of yarn in four randomly chosen balls of 3-ply yarn from Machine B.</p><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 company produces two types of yarn balls with two machines. Machine A produces balls of 2-ply yarn. Over a period of time, it is found that $80\%$ of the balls of yarn from Machine A have lengths less than $340$ metres and $10\%$ of the balls of yarn from Machine A have lengths less than $280$ metres.

Assuming that the lengths, in metres, of yarn in balls of 2-ply yarn from Machine A follows the distribution $N(\mu, \sigma^2)$, find $\mu$ and show that $\sigma = 28.3$.

For the rest of the question let $\mu = 300$ and $\sigma = 20$.

Machine B produces balls of 3-ply yarn. The lengths, in metres, of yarn in balls of 3-ply yarn from Machine B follows the distribution $N(220, 6^2)$.

Five balls of 3-ply yarn from Machine B are randomly chosen. Find the probability that two of them have lengths more than 230 m and three of them have lengths less than 215 m.

The probability that two randomly chosen balls of 2-ply yarn from Machine A and three randomly chosen balls of 3-ply yarn from Machine B have total length exceeding $k$ metres is 0.385. Find the value of $k$, correct to 2 decimal places.

2025 JPJC P2 Q10

Sampling

<p style="margin-bottom: 1rem">The weights of girls and boys in a college have independent normal distributions with their means and standard deviations given in the following table.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 450px;"><colgroup><col style="width: 150px;"><col style="width: 150px;"><col style="width: 150px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Mean</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Standard Deviation</p></th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Girls</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$44$ kg</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$5$ kg</p></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Boys</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$50$ kg</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$4$ kg</p></td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the greatest weight (correct to nearest $0.1$ kg) exceeded by at least $10\%$ of the boys.	<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the probability that the difference of the weights of $2$ randomly chosen girls is less than $1.2$ kg.	<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">One girl and one boy are chosen at random. Find the probability that one of them is at least $47$ kg and the other is 	less than $47$ kg.	<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the probability that the average of $2$ randomly chosen girls and $8$ randomly chosen boys is less than $50$ kg.<span class="inline-marks">[3]</span></div></li></ol>

The weights of girls and boys in a college have independent normal distributions with their means and standard deviations given in the following table.

Linear Combinations, Statistical Modeling, and Probabilistic Inference

<p>The lifespan of a halogen bulb is normally distributed with mean $160$ hours with standard deviation $10$ hours, while the lifespan of a fluorescent bulb is normally distributed with mean $240$ hours and standard deviation $12$ hours. The lifespan of any bulb is independent of one another.</p>
<ol class="i-list"><li><div class="list-item-paragraph">A halogen bulb is randomly chosen. Find the greatest value of $a$, correct to three significant figures, if the probability that its lifespan lies in the range of $(160-a,160+a)$ is at most $0.4$.	<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the probability that the difference between the average lifespan of two fluorescent bulbs and twice the lifespan of a halogen bulb does not exceed $70$ hours. 	<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">The halogen bulbs are packed in boxes of $n$ bulbs. If there are more than $10$ bulbs that have lifespans of less than $150$ hours, the box will be rejected. Find the greatest value of $n$ so that the probability that a box will be rejected is less than $0.2$. <span class="inline-marks">[5]</span></div></li></ol>

The lifespan of a halogen bulb is normally distributed with mean $160$ hours with standard deviation $10$ hours, while the lifespan of a fluorescent bulb is normally distributed with mean $240$ hours and standard deviation $12$ hours. The lifespan of any bulb is independent of one another.

<p>An orchard owned by Mark produces oranges which have masses (in grams) that follow a normal distribution $\text{N}\hspace{0.1em}\left( 190,576 \right)$. Visitors to this orchard can buy the oranges at $\$0.10$ per $100$ g.<br></p>
<p>Find the probability that the payment made by a visitor buying $20$ oranges will differ from $2$ times the payment made by another visitor buying $10$ oranges by at most $\$0.15$.	<span class="inline-marks">[4]</span></p>
<p>The orchard produces apples which are graded according to their mass. Apples with a mass exceeding $150$ g are graded as ‘large’ while apples with a mass less than $70$ g are graded as ‘small’. </p>
<p>Mark finds that the proportion of apples graded as ‘large’ is the same as that of the apples graded as ‘small’. It is given that the mass of a randomly chosen apple from Mark’s orchard follows a normal distribution with mean $\mu $ g and standard deviation $30$ g. </p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the value of $\mu $. 	<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">What is the probability that Mark will get at least $5$ apples graded as ‘large’ when he randomly selects $65$ apples 	from his orchard?	<span class="inline-marks">[3]</span></div></li></ol>

An orchard owned by Mark produces oranges which have masses (in grams) that follow a normal distribution $\text{N}\left( 190,576 \right)$. Visitors to this orchard can buy the oranges at $\$0.10$ per $100$ g.


Find the probability that the payment made by a visitor buying $20$ oranges will differ from $2$ times the payment made by another visitor buying $10$ oranges by at most $\$0.15$.	

The orchard produces apples which are graded according to their mass. Apples with a mass exceeding $150$ g are graded as ‘large’ while apples with a mass less than $70$ g are graded as ‘small’. 

Mark finds that the proportion of apples graded as ‘large’ is the same as that of the apples graded as ‘small’. It is given that the mass of a randomly chosen apple from Mark’s orchard follows a normal distribution with mean $\mu $ g and standard deviation $30$ g. 

2010 HCI P2 Q9

<p>Two friends June and Cindy always meet in the library to study together. Their arrival time to the library and length of stay in the library is independent of each other. June’s arrival time follows a normal distribution with mean arrival time at $1700$ h with a standard deviation of $30$ minutes and her stay in the library follows a normal distribution with mean $40$ minutes and standard deviation $15$ minutes. On the other hand, Cindy will reach the library every day at $1730$ h sharp and her stay in the library follows a normal distribution with mean $1$ hour and standard deviation $15$ minutes.</p>
<ol class="i-list"><li><div class="list-item-paragraph">On a randomly chosen day, calculate the following probabilities:</div><ol class="abc-list"><li><div class="list-item-paragraph">Cindy is in the library earlier than June. <span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">June leaves before Cindy arrives. 	         <span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-paragraph">Calculate the probability that the total time spent by June for three days in the library is more than three times the 	time spent by Cindy in a day.      <span class="inline-marks">[2]</span></div></li></ol>
<p>For a particular week, Cindy is not consistent in turning up at the library. The probability that Cindy will visit the library on any particular day is $0.7$. If she visits the library, she will turn up at $1730$ h sharp and her stay will follow the same distribution as before.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Show that the probability that June and Cindy will not meet in a particular day is $0.5705$. 	         <span class="inline-marks">[1]</span></div></li></ol>

Two friends June and Cindy always meet in the library to study together. Their arrival time to the library and length of stay in the library is independent of each other. June’s arrival time follows a normal distribution with mean arrival time at $1700$ h with a standard deviation of $30$ minutes and her stay in the library follows a normal distribution with mean $40$ minutes and standard deviation $15$ minutes. On the other hand, Cindy will reach the library every day at $1730$ h sharp and her stay in the library follows a normal distribution with mean $1$ hour and standard deviation $15$ minutes.

For a particular week, Cindy is not consistent in turning up at the library. The probability that Cindy will visit the library on any particular day is $0.7$. If she visits the library, she will turn up at $1730$ h sharp and her stay will follow the same distribution as before.

Advanced Operations, Statistical Modeling, and Probabilistic Inference

<p style="margin-bottom: 1.25rem">A school has a large number of girls. For each year-group it may be assumed that the heights of girls are normally distributed, with average heights and standard deviations as given in the following table.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 450px;"><colgroup><col style="width: 150px;"><col style="width: 150px;"><col style="width: 150px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Year-group</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Average height</p><p style="text-align: center;"></p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Standard deviation</p></th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$11$-year old girls</p><p style="text-align: left;"></p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p style="text-align: center;">$1.1$ m</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$0.2$ m</p></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$16$-year old girls</p><p style="text-align: left;"></p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p style="text-align: center;">$1.7$ m</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$0.3$ m</p></td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-paragraph">For each of the following cases, state, with a reason, whether or not a normal model is likely to be appropriate.</div><ol class="i-list"><li><div class="list-item-paragraph">The number of days since her last birthday of a randomly chosen $11$-year old girl from the school.</div></li><li><div class="list-item-paragraph">The height of a pupil chosen at random from the combined group of $11$-year old and $16$-year old girls in the school.<br></div></li><li><div class="list-item-paragraph">The average height of a randomly chosen sample of eight $11$-year old girls from the school</div></li></ol></li><li><div class="list-item-paragraph">Find the probability such that</div><ol class="i-list"><li><div class="list-item-paragraph">each of the two randomly chosen $11$-year old girls is taller than $1.2$ m.</div></li><li><div class="list-item-paragraph">the total height of two randomly chosen $11$-year old girls is more than $2.4$ m.</div></li></ol></li></ol>
<p>Explain why the answer to <strong>(b)(ii)</strong> is higher than that of <strong>(b)(i)</strong>.</p>

A school has a large number of girls. For each year-group it may be assumed that the heights of girls are normally distributed, with average heights and standard deviations as given in the following table.

<p>Average height</p><p style="text-align: center;"></p>

Explain why the answer to **(b)(ii)** is higher than that of **(b)(i)**.

Linear Combinations, Parameter Estimation, and Real-World Modeling

<p>Given two independent normal distributions $X\sim N(10,12)$ and $Y\sim N(5,8)$. Find the normal distribution of</p>
<ol class="i-list"><li><div class="list-item-paragraph">$4X+2Y$ </div></li><li><div class="list-item-paragraph">$3X-4Y$ </div></li><li><div class="list-item-paragraph">${{X}_{1}}+{{X}_{2}}+{{X}_{3}}-{{Y}_{1}}-{{Y}_{2}}-{{Y}_{3}}-{{Y}_{4}}$ </div></li></ol>

Given two independent normal distributions $X\sim N(10,12)$ and $Y\sim N(5,8)$. Find the normal distribution of

Linear Combinations of Normal Random Variables

<p>It is given that $X\sim N(\mu ,{{\sigma }^{2}})$ and$\mathrm{P}(X&lt;9)=\mathrm{P}(X&gt;6)$ . It’s also given that $5\mathrm{P}(X&lt;4)=2\mathrm{P}(X&lt;11)$. Find $\mu $ and $\sigma $.</p>

It is given that $X\sim N(\mu ,{{\sigma }^{2}})$ and$\mathrm{P}(X<9)=\mathrm{P}(X>6)$ . It’s also given that $5\mathrm{P}(X<4)=2\mathrm{P}(X<11)$. Find $\mu $ and $\sigma $.

Symmetrical Properties Of Normal Distribution Q6

<p>It is given that $X\sim N(\mu ,35)$ and $P(X&lt;k)=P(X&gt;7)$. </p>
<ol class="i-list"><li><div class="list-item-paragraph">Express $k$ in terms of $\mu $.</div></li></ol>
<p>Given that&nbsp;$P(X&lt;5.4)=0.644$,</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">find the value of $k$. </div></li></ol>

It is given that $X\sim N(\mu ,35)$ and $P(X<k)=P(X>7)$. 

Symmetrical Properties Of Normal Distribution Q3

<p>Given that&nbsp;$X\sim N(\mu ,{{\sigma }^{2}})$,  $\text{P}\hspace{0.1em}(X&gt;25)=0.269$ and $\text{P}\hspace{0.1em}(X&gt;2)=0.783$. Find $\mu $ and $\sigma $. </p>

Given that $X\sim N(\mu ,{{\sigma }^{2}})$,  $\text{P}(X>25)=0.269$ and $\text{P}(X>2)=0.783$. Find $\mu $ and $\sigma $. 

Standard Normal Distribution 

<p>Given that&nbsp;$X\sim N(-34,70)$ and $\text{P}\hspace{0.1em}(\left| X-c \right|&gt;13)=0.257$. Find $c$.</p>

Given that $X\sim N(-34,70)$ and $\text{P}(\left| X-c \right|>13)=0.257$. Find $c$.

Basics of Normal Distribution using Calculators Q18

<p>Given that&nbsp;$X\sim N(-20,83)$ and $\text{P}\hspace{0.1em}(\left| X-a \right|&gt;2.1)=0.85$. Find $a$.</p>

Given that $X\sim N(-20,83)$ and $\text{P}(\left| X-a \right|>2.1)=0.85$. Find $a$.

Math Question Bank