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<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 $0.95$.

Finding Minimum of Number of Trials

Basics of Binomial Distribution using Calculator

Problems Involving Binomial Distribution

Binomial Distribution

<p>A bag contains $2$ red balls, $3$ yellow balls and $1$ blue ball. Temmy and Ryan play a game where each takes turns to draw a ball at random from the bag, with replacement. The number of red balls obtained in $n$ fixed draws from the bag is denoted by $R$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">State, in context, an assumption satisfied by $R$ for it to be well modelled by a binomial distribution.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Temmy and Ryan each draw $n$ times from the bag. Find the least $n$ such that the probability of both getting a total of at most $10$ red balls is not more than $0.5$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Temmy and Ryan each draw $5$ times from the bag. The player with more red balls drawn wins. Otherwise, the game ends in a draw. Find the probability that Temmy wins the game if she draws more than $3$ red balls.</span></div></li></ol>
<p>In a variation of the game, Temmy draws balls at random from the bag, one at a time without replacement, and stops when she obtains $2$ yellow balls. The total number of balls Temmy has to draw from the bag before she stops is denoted by $T$. Find $\mathrm{E}(T)$ and $ \mathrm{Var}(T)$.</p>

A bag contains $2$ red balls, $3$ yellow balls and $1$ blue ball. Temmy and Ryan play a game where each takes turns to draw a ball at random from the bag, with replacement. The number of red balls obtained in $n$ fixed draws from the bag is denoted by $R$.

State, in context, an assumption satisfied by $R$ for it to be well modelled by a binomial distribution.

Temmy and Ryan each draw $n$ times from the bag. Find the least $n$ such that the probability of both getting a total of at most $10$ red balls is not more than $0.5$.

Temmy and Ryan each draw $5$ times from the bag. The player with more red balls drawn wins. Otherwise, the game ends in a draw. Find the probability that Temmy wins the game if she draws more than $3$ red balls.

In a variation of the game, Temmy draws balls at random from the bag, one at a time without replacement, and stops when she obtains $2$ yellow balls. The total number of balls Temmy has to draw from the bag before she stops is denoted by $T$. Find $\mathrm{E}(T)$ and $ \mathrm{Var}(T)$.

<p>In the mass production of bolts, it is found that&nbsp;$5$% are defective. Bolts are selected at random and put into packets of ten.</p>
<p>A packet is sold at reduced price if more than $1$ bolt is defective.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that a randomly selected packet of bolts is sold at reduced price.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that two randomly chosen packets each will be sold at reduced price.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>Last Monday, $20$ packets of bolts were sold.</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the probability that fewer than $3$ of these packets were sold at reduced price.</span><span class="inline-marks">[2]</span></div></li></ol>

In the mass production of bolts, it is found that $5$% are defective. Bolts are selected at random and put into packets of ten.

A packet is sold at reduced price if more than $1$ bolt is defective.

Find the probability that a randomly selected packet of bolts is sold at reduced price.

Find the probability that two randomly chosen packets each will be sold at reduced price.

Last Monday, $20$ packets of bolts were sold.

Find the probability that fewer than $3$ of these packets were sold at reduced price.

2024 ASRJC H1 Q6

<p>A shop sells plums in bags of $9$. Plums are taken from a large supply and the probability that a plum is rotten is $p$, where&nbsp;$0 &lt; p &lt; 0.2$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">It is known that the variance of the number of rotten plums in a bag is $0.6624$. Find $p$.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>It is now given that&nbsp;$p = 0.08$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the probability that a randomly selected bag of plums has at least $3$ rotten plums.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Five bags of plums are randomly selected. Find the probability that the fifth selected bag is the second bag with at least $3$ rotten plums.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">The shop sells $30$ bags of plums to a customer. Find the probability that less than $90$% of the bags have fewer than $3$ rotten plums.</span><span class="inline-marks">[3]</span></div></li></ol>

A shop sells plums in bags of $9$. Plums are taken from a large supply and the probability that a plum is rotten is $p$, where $0 < p < 0.2$.

It is known that the variance of the number of rotten plums in a bag is $0.6624$. Find $p$.

Find the probability that a randomly selected bag of plums has at least $3$ rotten plums.

Five bags of plums are randomly selected. Find the probability that the fifth selected bag is the second bag with at least $3$ rotten plums.

The shop sells $30$ bags of plums to a customer. Find the probability that less than $90$% of the bags have fewer than $3$ rotten plums.

2024 YIJC H1 Q9

<p>In a large retail company, the probability of a telesales representative making a sale on a customer call is constant at $0.15$ for every call made.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that, in $25$ random calls, at least $7$ but fewer than $13$ sales are made.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">If the probability that, in $100$ random calls, more than $r$ sales were made is less than $0.5$, find the least value of $r$.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>Telesales representatives are required to achieve an average of at least $5$ sales per day.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the least number of calls each day a representative should make to meet this requirement.</span><span class="inline-marks">[2]</span></div></li></ol>

In a large retail company, the probability of a telesales representative making a sale on a customer call is constant at $0.15$ for every call made.

Find the probability that, in $25$ random calls, at least $7$ but fewer than $13$ sales are made.

If the probability that, in $100$ random calls, more than $r$ sales were made is less than $0.5$, find the least value of $r$.

Telesales representatives are required to achieve an average of at least $5$ sales per day.

Find the least number of calls each day a representative should make to meet this requirement.

2024 VJC H1 Q10

<p>A fruits company sells mangoes in bags of $10$. The $10$ mangoes are randomly chosen from a large supply. The probability that a mango is rotten is $0.12$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that, in a randomly selected bag of mangoes, exactly one mango is rotten.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>A bag of mangoes is sold at a discounted price if more than $2$ mangoes in the bag are rotten.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the probability that a randomly selected bag of mangoes is sold at the discounted price.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>Two bags of mangoes are selected at random.</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the probability that both bags are sold at its original price.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>Last Sunday, $50$ bags of mangoes were sold.</p>
<ol class="i-list" start="4"><li><div class="list-item-content"><span class="list-item-text">Find the probability that fewer than $3$ of these bags were sold at the discounted price.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>The company has a wholesaler subsidiary and sells to fruit sellers in cartons of $n$ mangoes, where&nbsp;$n$ is greater than $10$. Find the least number of mangoes needed to be packed in such cartons so that there is a probability of less than $0.15$ that no more than $5$ rotten mangoes are found in a carton.<span class="inline-marks">[2]</span></p>

A fruits company sells mangoes in bags of $10$. The $10$ mangoes are randomly chosen from a large supply. The probability that a mango is rotten is $0.12$.

Find the probability that, in a randomly selected bag of mangoes, exactly one mango is rotten.

A bag of mangoes is sold at a discounted price if more than $2$ mangoes in the bag are rotten.

Find the probability that a randomly selected bag of mangoes is sold at the discounted price.

Two bags of mangoes are selected at random.

Find the probability that both bags are sold at its original price.

Last Sunday, $50$ bags of mangoes were sold.

Find the probability that fewer than $3$ of these bags were sold at the discounted price.

The company has a wholesaler subsidiary and sells to fruit sellers in cartons of $n$ mangoes, where $n$ is greater than $10$. Find the least number of mangoes needed to be packed in such cartons so that there is a probability of less than $0.15$ that no more than $5$ rotten mangoes are found in a carton.

2024 RVHS H1 Q9

<p>In a large population, the proportion of people who wear glasses is $62.1$%. A group of $9$ people is randomly chosen from the population.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that exactly $3$ of these people wear glasses.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that at least five of these people wear glasses.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>The health promotion board decides to study $10$ of such groups chosen randomly from the population.</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the probability that less than half of the groups have at least five people wearing glasses.</span><span class="inline-marks">[3]</span></div></li></ol>

In a large population, the proportion of people who wear glasses is $62.1$%. A group of $9$ people is randomly chosen from the population.

Find the probability that exactly $3$ of these people wear glasses.

Find the probability that at least five of these people wear glasses.

The health promotion board decides to study $10$ of such groups chosen randomly from the population.

Find the probability that less than half of the groups have at least five people wearing glasses.

2024 MI H1 Q7

<p>A manufacturer produces mystery boxes, each containing either a regular or a seasonal toy. On average, $p$% of the mystery boxes contain a seasonal toy.</p>
<p>Amber orders $n$ mystery boxes from the manufacturer. The number of seasonal toys that Amber gets is the random variable $S$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">State, in the context of the question, two assumptions needed for&nbsp;$S$ to be well modelled by a binomial distribution.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>You are now given that&nbsp;$S$ can be modelled by a binomial distribution.</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$\mathrm{P}(S=2)=\mathrm{P}(S=3)$ and $\mathrm{E}(S)=2.96$, find the value of $p$.</span><span class="inline-marks">[4]</span></div></li></ol>
<p>Assume now that&nbsp;$p=5$.</p>
<p>The manufacturer now packs the mystery boxes into cartons of $12$ each for sale. Each carton is checked for quality control. If there is at most $1$ mystery box containing a seasonal toy, the carton is accepted. Otherwise, the carton is rejected.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Given that a randomly chosen carton is rejected, find the probability that no more than $30$% of the boxes in the carton each contains a seasonal toy.</span><span class="inline-marks">[3]</span></div></li></ol>

A manufacturer produces mystery boxes, each containing either a regular or a seasonal toy. On average, $p$% of the mystery boxes contain a seasonal toy.

Amber orders $n$ mystery boxes from the manufacturer. The number of seasonal toys that Amber gets is the random variable $S$.

State, in the context of the question, two assumptions needed for $S$ to be well modelled by a binomial distribution.

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

Given that $\mathrm{P}(S=2)=\mathrm{P}(S=3)$ and $\mathrm{E}(S)=2.96$, find the value of $p$.

The manufacturer now packs the mystery boxes into cartons of $12$ each for sale. Each carton is checked for quality control. If there is at most $1$ mystery box containing a seasonal toy, the carton is accepted. Otherwise, the carton is rejected.

Given that a randomly chosen carton is rejected, find the probability that no more than $30$% of the boxes in the carton each contains a seasonal toy.

2025 TMJC P2 Q8

<p>A bag contains $4$ identical red balls and $6$ identical blue balls.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">In the first game, the balls are randomly picked by a player from the bag, one at a time, without replacement, until there are no balls left in the bag.</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that the last ball picked is red.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that in the first five picks, exactly $2$ blue balls are picked given that at least $3$ red balls are picked.</span><span class="inline-marks">[3]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">In the second game, each ball picked by a player will be placed back into the bag before the next ball is picked. The player makes a total of $20$ random picks from the bag and the colour of each pick is recorded.</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that the colour red is recorded exactly $4$ times.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that the colour red is recorded more than $4$ times but not more than $8$ times.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that the $8$th pick is the $6$th time the colour blue is recorded.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">The second game is played by $50$ randomly chosen players. Estimate the probability that the average number of times the colour red is recorded is more than $8.5$.</span><span class="inline-marks">[4]</span></div></li></ol></li></ol>

A bag contains $4$ identical red balls and $6$ identical blue balls.

In the first game, the balls are randomly picked by a player from the bag, one at a time, without replacement, until there are no balls left in the bag.

In the second game, each ball picked by a player will be placed back into the bag before the next ball is picked. The player makes a total of $20$ random picks from the bag and the colour of each pick is recorded.

2025 RI P2 Q10

Probability

<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 $9$.

Binomial Distribution - General

2015 TJC P2 Q9 [Modified]

<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, $27.5$% of the population has dimples on their cheeks. Samples of $12$ people are randomly chosen from the population.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Calculate the probability that more than $4$ people in a sample has dimples on their cheeks.</span></div></li><li><div class="list-item-content"><span class="list-item-text">$4$ such samples of $12$ people are taken. Find the probability that</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Each of the $4$ samples has more than $4$ people who have dimples on their cheeks.</span></div></li><li><div class="list-item-content"><span class="list-item-text">One of the $4$ samples has exactly $3$ persons who have dimples on their cheeks, while the remaining $3$ samples have more than $3$ people who have dimples on their cheeks.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Two of the $4$ samples have less than $4$ persons who have dimples on their cheeks, another has exactly $4$ persons who have dimples on their cheeks and the remaining sample has more than $4$ persons who have dimples on their cheeks.</span></div></li><li><div class="list-item-content"><span class="list-item-text">There are total of $8$ people who have dimples on their cheeks from the $4$ samples.</span></div></li></ol></li></ol>

In a large population, $27.5$% of the population has dimples on their cheeks. Samples of $12$ people are randomly chosen from the population.

Calculate the probability that more than $4$ people in a sample has dimples on their cheeks.

$4$ such samples of $12$ people are taken. Find the probability that

<p>Given that&nbsp;$X\sim\mathrm{B}(7,p)$ and $\mathrm{P}(X=4)=0.26102$. Find the possible values of $p$.</p>

Given that $X\sim\mathrm{B}(7,p)$ and $\mathrm{P}(X=4)=0.26102$. Find the possible values of $p$.

<p>Given that&nbsp;$X\sim\mathrm{B}(50,p)$ and $\mathrm{Var}(X)=5.32$. Find the possible values of $p$.</p>

Given that $X\sim\mathrm{B}(50,p)$ and $\mathrm{Var}(X)=5.32$. Find the possible values of $p$.

<p>Given that&nbsp;$X\sim\mathrm{B}(45,p)$ and $\mathrm{E}(X)=10$. Find $p$.</p>

Given that $X\sim\mathrm{B}(45,p)$ and $\mathrm{E}(X)=10$. Find $p$.

<p>Given that&nbsp;$X \sim \mathrm{B}(40, 0.64)$, find</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(X=28)$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(X &gt; 21)$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(23.3 \le X \le 31.7)$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(X &gt; 28.6)$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(|X - 31| &lt; 4)$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(|X - 26| \ge 4)$</span></div></li><li><div class="list-item-content"><span class="list-item-text">Probability that&nbsp;$X$ lies within two standard deviations from the mean.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Mode of $X$.</span></div></li></ol>

Given that $X \sim \mathrm{B}(40, 0.64)$, find

Probability that $X$ lies within two standard deviations from the mean.

<p>Given that&nbsp;$X \sim \mathrm{B}(35, 0.43)$, find</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(X=17)$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(X &gt; 14)$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(7.3 \le X &lt; 12.1)$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(X \ge 18.2)$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(|X - 13| &lt; 2)$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(|X - 12| &gt; 5)$</span></div></li><li><div class="list-item-content"><span class="list-item-text">Probability that&nbsp;$X$ lies within one standard deviation from the mean.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Mode of $X$.</span></div></li></ol>

Given that $X \sim \mathrm{B}(35, 0.43)$, find

Probability that $X$ lies within one standard deviation from the mean.

<p>Given that&nbsp;$X \sim \mathrm{B}(5, 0.67)$, without using the GC, find</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(X=2)$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(X &lt; 2)$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(1 \le X \le 3)$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(X &lt; 4)$</span></div></li></ol>

Given that $X \sim \mathrm{B}(5, 0.67)$, without using the GC, find

<p>$65\%$ of the customers in a queue to a PC fair are males. </p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the probability that there first three customers are females and next three are males.</div></li><li><div class="list-item-paragraph">Find the probability that the $4^{\mathrm{th}}$ male customer is the $7^{\mathrm{th}}$ customer in the queue.</div></li><li><div class="list-item-paragraph">There are $20$ customers in the queue. Find probability that there are more than $9$ male customers and the $4^{\mathrm{th}}$ male customer is the $7^{\mathrm{th}}$ customer in the queue.</div></li></ol>
<p>There are $n$ customers waiting in another queue in the same supermarket. The probability that at least one customer is a female is greater than $0.995$. Find the minimum value of $n$.</p>

$65\%$ of the customers in a queue to a PC fair are males. 

There are $n$ customers waiting in another queue in the same supermarket. The probability that at least one customer is a female is greater than $0.995$. Find the minimum value of $n$.

<p>Customers wait in a queue in a supermarket. A customer can choose to pay with cash or a credit card. Assume that whether or not a customer pays with a credit card is independent of any other customers’ methods of payment. It is known that $60\%$ of customers choose to pay with a credit card.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the probability that the first three customers pay with a credit card and the next three pay with cash;</div></li><li><div class="list-item-paragraph">Find the probability that the $4^{\mathrm{th}}$ person who pays with credit card is the $6^{\mathrm{th}}$ customer in the queue.</div></li><li><div class="list-item-paragraph">There are $15$ customers in the queue. Find the probability that there are more than $10$ customers who pay with a credit card and the $4^{\mathrm{th}}$ person who pays with credit card is the $6^{\mathrm{th}}$ customer in the queue.</div></li></ol>
<p>There are $n$ customers waiting in another queue in the same supermarket. The probability that at least one customer pays with cash is greater than $0.995$.<br></p>
<p>Find the minimum value of $n$.</p>

Customers wait in a queue in a supermarket. A customer can choose to pay with cash or a credit card. Assume that whether or not a customer pays with a credit card is independent of any other customers’ methods of payment. It is known that $60\%$ of customers choose to pay with a credit card.

There are $n$ customers waiting in another queue in the same supermarket. The probability that at least one customer pays with cash is greater than $0.995$.


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