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<p>One round of a navigation game is played by moving a token from the starting point $P$ to one of the finishing points on the right-hand edge of the grid shown.</p>
<ul class="bullet-list"><li data-list-item-id="rule-1" data-list-item-path="rule-1"><div class="list-item-paragraph">The token may only move one grid spacing to the right or one grid spacing upwards at each move.</div></li><li data-list-item-id="rule-2" data-list-item-path="rule-2"><div class="list-item-paragraph">At every move after $P$, the token moves to the right with probability $p$ or upwards with probability $q$, where&nbsp;$p+q=1$. The moves are independent.</div></li><li data-list-item-id="rule-3" data-list-item-path="rule-3"><div class="list-item-paragraph">To win a round, the token must pass through $C$ and finish at $T$.</div></li></ul>
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<ol class="abc-list"><li data-list-item-id="part-a-item" data-list-item-path="part-a-item"><div class="list-item-paragraph">Show that the probability of winning a round is $45p^6q^3$.<span class="inline-marks">[3]</span></div></li></ol>
<p>It is given that&nbsp;$p=\frac34$. Aisha plays one game comprising 16 rounds. Each round is independent of the others. Let&nbsp;$W$ be the number of rounds that Aisha wins in one game.</p>
<ol class="abc-list" start="2"><li data-list-item-id="part-b-item" data-list-item-path="part-b-item"><div class="list-item-paragraph">Find the probability that Aisha wins at least 4 of the 16 rounds.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="part-c-item" data-list-item-path="part-c-item"><div class="list-item-paragraph"><p>Aisha plays 50 such games independently. Estimate the probability that the mean number of rounds she wins per game is at most 1.8. Give your answer to 3 significant figures.</p><span class="inline-marks">[4]</span></div></li></ol>

One round of a navigation game is played by moving a token from the starting point $P$ to one of the finishing points on the right-hand edge of the grid shown.

The token may only move one grid spacing to the right or one grid spacing upwards at each move.

At every move after $P$, the token moves to the right with probability $p$ or upwards with probability $q$, where $p+q=1$. The moves are independent.

To win a round, the token must pass through $C$ and finish at $T$.

Show that the probability of winning a round is $45p^6q^3$.

It is given that $p=\frac34$. Aisha plays one game comprising 16 rounds. Each round is independent of the others. Let $W$ be the number of rounds that Aisha wins in one game.

Find the probability that Aisha wins at least 4 of the 16 rounds.

Central Limit Theorem

Problems Involving Binomial Distribution

Navigation game through a checkpoint

Binomial Distribution

Sampling

<p>The random variable $X$ has the binomial distribution $\mathrm{B}(n,p)$, where&nbsp;$n\geq2$ and $0&lt;p&lt;1$. The mean of $X$ is three times its standard deviation.</p>
<ol class="abc-list"><li data-list-item-id="bimodal-mean-sd-q2-a" data-list-item-path="bimodal-mean-sd-q2-a"><div class="list-item-paragraph">Show that&nbsp;$p=\frac{9}{n+9}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="bimodal-mean-sd-q2-b" data-list-item-path="bimodal-mean-sd-q2-b"><div class="list-item-paragraph">Given that&nbsp;$X$ is bimodal and $n&lt;40$, find all possible values of $n$ and the corresponding modes of $X$.<span class="inline-marks">[4]</span></div></li></ol>

The random variable $X$ has the binomial distribution $\mathrm{B}(n,p)$, where $n\geq2$ and $0<p<1$. The mean of $X$ is three times its standard deviation.

Binomial Distribution - General

Bimodal Distribution

Two modes under a mean-standard-deviation condition

<p>Bluewater Ferries operates a service with $180$ passenger seats. To reduce the number of empty seats, the company may accept more reservations than the number of seats available.</p>
<p>Each reservation costs $\$72$ and is non-refundable. Past records show that a reservation holder does not attend with probability $0.07$. If more than $180$ reservation holders attend, the company pays $\$320$ in compensation to each passenger who is unable to board.</p>
<p>Assume that reservation holders attend independently and that the attendance probability is the same for every reservation. For a particular crossing, the company accepts $190$ reservations. Let&nbsp;$X$ denote the number of reservation holders who attend.</p>
<ol class="abc-list"><li data-list-item-id="q2-a" data-list-item-path="q2-a"><div class="list-item-paragraph">Find the probability that there are insufficient seats for the crossing.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="q2-b" data-list-item-path="q2-b"><div class="list-item-paragraph">Calculate the expected net revenue from the crossing, giving your answer to the nearest dollar.<span class="inline-marks">[3]</span></div></li></ol>
<p>Suppose instead that the company accepts $n$ reservations, where&nbsp;$n&gt;180$, and let&nbsp;$X\sim\mathrm{B}(n,0.93)$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="q2-c" data-list-item-path="q2-c"><div class="list-item-paragraph">Write down an expression, in terms of $n$ and binomial probabilities, for the expected net revenue from one crossing.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="q2-d" data-list-item-path="q2-d"><div class="list-item-paragraph">Hence, determine the number of reservations that the company should accept to maximise the expected net revenue, and state this maximum expected net revenue to the nearest dollar.<span class="inline-marks">[3]</span></div></li></ol>

Bluewater Ferries operates a service with $180$ passenger seats. To reduce the number of empty seats, the company may accept more reservations than the number of seats available.

Each reservation costs $\$72$ and is non-refundable. Past records show that a reservation holder does not attend with probability $0.07$. If more than $180$ reservation holders attend, the company pays $\$320$ in compensation to each passenger who is unable to board.

Assume that reservation holders attend independently and that the attendance probability is the same for every reservation. For a particular crossing, the company accepts $190$ reservations. Let $X$ denote the number of reservation holders who attend.

Suppose instead that the company accepts $n$ reservations, where $n>180$, and let $X\sim\mathrm{B}(n,0.93)$.

Ferry overbooking and expected net revenue

<p>The random variable $X$ has the binomial distribution $\mathrm{B}(n,p)$, where&nbsp;$n$ is a positive integer and $0&lt;p&lt;1$. Denote $\mathrm{P}(X=r)$ by $q_r$.</p>
<ol class="abc-list"><li data-list-item-id="bimodal-mean-q1-a" data-list-item-path="bimodal-mean-q1-a"><div class="list-item-paragraph">Show that&nbsp;$\frac{q_r}{q_{r-1}}=\frac{(n-r+1)p}{r(1-p)}$, for&nbsp;$r=1,2,\ldots,n$. Hence show that&nbsp;$X$ is bimodal if and only if&nbsp;$(n+1)p$ is an integer, and state its two modes in terms of $n$ and $p$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="bimodal-mean-q1-b" data-list-item-path="bimodal-mean-q1-b"><div class="list-item-paragraph">It is known that the modes of $X$ are $11$ and $12$, and that&nbsp;$\mathrm{E}(X)=11.52$. Determine $n$ and $p$.<span class="inline-marks">[3]</span></div></li></ol>

The random variable $X$ has the binomial distribution $\mathrm{B}(n,p)$, where $n$ is a positive integer and $0<p<1$. Denote $\mathrm{P}(X=r)$ by $q_r$.

Two modes from a probability ratio and mean

<p>The Horizon Planetarium has&nbsp;$96$ seats for each evening show. To reduce the number of empty seats, it accepts more reservations than the number of seats available. Past records show that a reservation holder attends the show with probability $0.88$. For a particular evening show, $106$ reservations have been accepted. Let&nbsp;$X$ denote the number of reservation holders who attend.</p>
<ol class="abc-list"><li data-list-item-id="q1-a" data-list-item-path="q1-a"><div class="list-item-paragraph">State two assumptions needed for&nbsp;$X$ to be well modelled by a binomial distribution.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="q1-b" data-list-item-path="q1-b"><div class="list-item-paragraph">Assume that the assumptions in part <strong>(a)</strong> hold. Find the probability that there are insufficient seats for the show.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="q1-c" data-list-item-path="q1-c"><div class="list-item-paragraph">Determine the maximum number of reservations that the planetarium may accept so that the probability of having insufficient seats is less than $2\%$.<span class="inline-marks">[3]</span></div></li></ol>

The Horizon Planetarium has $96$ seats for each evening show. To reduce the number of empty seats, it accepts more reservations than the number of seats available. Past records show that a reservation holder attends the show with probability $0.88$. For a particular evening show, $106$ reservations have been accepted. Let $X$ denote the number of reservation holders who attend.

Planetarium reservation limit under a risk constraint

<p>Two brothers Kai and Leo are duathlon (running and cycling) athletes who train regularly. For each day, the probability that Kai cycles is $\frac{3}{4}$, the probability that he runs is $\frac{3}{5}$ and the probability he does both is $p$.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-1-item-0" data-list-item-path="imported-item-block-1-item-0"><div class="list-item-paragraph">Write down, in terms of $p$, the probability that, on one day, Kai either runs or cycles but not both.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-1-item-1" data-list-item-path="imported-item-block-1-item-1"><div class="list-item-paragraph">Find the range of possible values of $p$.<span class="inline-marks">[2]</span></div></li></ol>
<p>On average, Leo cycles $5$ out of $7$ days in a week. The probability that Leo cycles when Kai cycles is $0.9$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="imported-item-block-3-item-0" data-list-item-path="imported-item-block-3-item-0"><div class="list-item-paragraph">Find the probability that Leo cycles when Kai does not.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-1" data-list-item-path="imported-item-block-3-item-1"><div class="list-item-paragraph">State, in context, two assumptions needed for the number of days Leo cycles over a period of $5$ weeks to be well modelled by a binomial distribution.<span class="inline-marks">[2]</span></div></li></ol>
<p>Assume now that the number of days Leo cycles in $5$ weeks has a binomial distribution.</p>
<ol class="abc-list" start="5"><li data-list-item-id="imported-item-block-5-item-0" data-list-item-path="imported-item-block-5-item-0"><div class="list-item-paragraph">Find the probability that, in the $5$ weeks, Leo cycles on at least $20$ days but fewer than $30$ days.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-5-item-1" data-list-item-path="imported-item-block-5-item-1"><div class="list-item-paragraph">Find the probability that, in the $5$ weeks, there are exactly $5$ days in which both brothers do not cycle and they both cycle on the other days.<span class="inline-marks">[2]</span></div></li></ol>

Two brothers Kai and Leo are duathlon (running and cycling) athletes who train regularly. For each day, the probability that Kai cycles is $\frac{3}{4}$, the probability that he runs is $\frac{3}{5}$ and the probability he does both is $p$.

Write down, in terms of $p$, the probability that, on one day, Kai either runs or cycles but not both.

Find the range of possible values of $p$.

On average, Leo cycles $5$ out of $7$ days in a week. The probability that Leo cycles when Kai cycles is $0.9$.

Find the probability that Leo cycles when Kai does not.

State, in context, two assumptions needed for the number of days Leo cycles over a period of $5$ weeks to be well modelled by a binomial distribution.

Assume now that the number of days Leo cycles in $5$ weeks has a binomial distribution.

Find the probability that, in the $5$ weeks, Leo cycles on at least $20$ days but fewer than $30$ days.

Find the probability that, in the $5$ weeks, there are exactly $5$ days in which both brothers do not cycle and they both cycle on the other days.

2024 RI P2 Q11

<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

<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 factory produces porcelain bowls. It is known that, on average, $8\%$ of the bowls are faulty. The bowls are packed in boxes of $20$. A box is considered imperfect if it contains more than $2$ faulty bowls. Assume that the number of faulty bowls in a box follows a binomial distribution.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the probability that a randomly chosen box is imperfect.<span class="inline-marks">[2]</span></div></li></ol>
<p>These boxes are packed into cartons of $12$ boxes each.</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 probability that, out of $3$ randomly chosen cartons, there are $2$ cartons that contains fewer than $15$ faulty bowls each.<span class="inline-marks">[3]</span></div></li></ol>
<p>Let&nbsp;$Y$ be the number of imperfect boxes in a randomly chosen carton.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">State the values of $\mathrm{E}(Y)$ and $\mathrm{Var}(Y)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Hence using a suitable approximation, find the probability that the total number of imperfect boxes in a random sample of $35$ cartons is at most $85$.<span class="inline-marks">[2]</span></div></li></ol>

A factory produces porcelain bowls. It is known that, on average, $8\%$ of the bowls are faulty. The bowls are packed in boxes of $20$. A box is considered imperfect if it contains more than $2$ faulty bowls. Assume that the number of faulty bowls in a box follows a binomial distribution.

Find the probability that a randomly chosen box is imperfect.

These boxes are packed into cartons of $12$ boxes each.

Find the probability that, out of $3$ randomly chosen cartons, there are $2$ cartons that contains fewer than $15$ faulty bowls each.

Let $Y$ be the number of imperfect boxes in a randomly chosen carton.

State the values of $\mathrm{E}(Y)$ and $\mathrm{Var}(Y)$.

Hence using a suitable approximation, find the probability that the total number of imperfect boxes in a random sample of $35$ cartons is at most $85$.

2025 TJC P2 Q7

<p>A bag contains $4$ identical red balls and $6$ identical blue balls.</p>
<ol class="abc-list"><li data-list-item-id="item-a-5" data-list-item-path="item-a-5"><div class="list-item-paragraph">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.</div><ol class="i-list"><li data-list-item-id="item-a-5-i" data-list-item-path="item-a-5/item-a-5-i"><div class="list-item-paragraph">Find the probability that the last ball picked is red.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-a-5-ii" data-list-item-path="item-a-5/item-a-5-ii"><div class="list-item-paragraph">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 class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-b-5" data-list-item-path="item-b-5"><div class="list-item-paragraph">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.</div><ol class="i-list"><li data-list-item-id="item-b-5-i" data-list-item-path="item-b-5/item-b-5-i"><div class="list-item-paragraph">Find the probability that the colour red is recorded exactly $4$ times.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-5-ii" data-list-item-path="item-b-5/item-b-5-ii"><div class="list-item-paragraph">Find the probability that the colour red is recorded more than $4$ times but not more than $8$ times.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-5-iii" data-list-item-path="item-b-5/item-b-5-iii"><div class="list-item-paragraph"><p>Find the probability that the $8 ^\mathrm{th}$ pick is the $6^\mathrm{th}$ time the colour blue is recorded.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-5-iv" data-list-item-path="item-b-5/item-b-5-iv"><div class="list-item-paragraph">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 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>A small ceramic workshop produces <strong>100</strong> plates each working day. Some of the plates turn out to be faulty.</p>
<ol class="abc-list"><li data-list-item-id="item-5-a" data-list-item-path="item-5-a"><div class="list-item-paragraph">State, in the context of the question, two assumptions needed for the number of faulty plates made in a day to be well modelled by a binomial distribution.<span class="inline-marks">[2]</span></div></li></ol>
<p>Assume now that the number of faulty plates produced each working day has the distribution $\mathrm{B}(100, p)$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-5-b" data-list-item-path="item-5-b"><div class="list-item-paragraph">Show that the probability that exactly $3$ faulty plates are produced on a randomly chosen working day is $161700p^3(1-p)^{97}$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-5-c" data-list-item-path="item-5-c"><div class="list-item-paragraph">Given that the most likely number of faulty plates produced on a working day is $3$, find the possible range of values of $p$, leaving your answer in exact form.<span class="inline-marks">[4]</span></div></li></ol>
<p>The workshop also produces bowls on each working day. The number of faulty bowls also follows a binomial distribution. The probability that a bowl is faulty is $q$. Faults on plates are independent of faults on bowls. The plates and bowls are sold in sets of $2$ randomly chosen bowls and $2$ randomly chosen plates. In the case where&nbsp;$p = 0.01$, the probability that a set contains at most $1$ faulty item is $0.88$.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-5-d" data-list-item-path="item-5-d"><div class="list-item-paragraph">Write down an equation satisfied by $q$. Hence find the value of $q$.<span class="inline-marks">[4]</span></div></li></ol>

A small ceramic workshop produces **100** plates each working day. Some of the plates turn out to be faulty.

State, in the context of the question, two assumptions needed for the number of faulty plates made in a day to be well modelled by a binomial distribution.

Assume now that the number of faulty plates produced each working day has the distribution $\mathrm{B}(100, p)$.

Show that the probability that exactly $3$ faulty plates are produced on a randomly chosen working day is $161700p^3(1-p)^{97}$.

Given that the most likely number of faulty plates produced on a working day is $3$, find the possible range of values of $p$, leaving your answer in exact form.

The workshop also produces bowls on each working day. The number of faulty bowls also follows a binomial distribution. The probability that a bowl is faulty is $q$. Faults on plates are independent of faults on bowls. The plates and bowls are sold in sets of $2$ randomly chosen bowls and $2$ randomly chosen plates. In the case where $p = 0.01$, the probability that a set contains at most $1$ faulty item is $0.88$.

Write down an equation satisfied by $q$. Hence find the value of $q$.

2025 NYJC P2 Q9

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

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

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

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

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

2013 DHS P2 Q10

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">The random variable $Y$ has a binomial distribution with mean $1.6$ and $\mathrm{P}(Y=0)=0.1296$. Find $\mathrm{P}(Y&gt;2)$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">A car park has a large number of parking lots. $80$ parking lots are observed and the number of occupied parking lots is denoted by $X$. State, in context, two assumptions needed for&nbsp;$X$ to be well modelled by a binomial distribution.</div></li></ol>
<p>The probability of a parking lot being occupied is $0.95$. Find the probability that out of the $80$ chosen parking lots, at least $90$% are occupied.</p>

The random variable $Y$ has a binomial distribution with mean $1.6$ and $\mathrm{P}(Y=0)=0.1296$. Find $\mathrm{P}(Y>2)$.

A car park has a large number of parking lots. $80$ parking lots are observed and the number of occupied parking lots is denoted by $X$. State, in context, two assumptions needed for $X$ to be well modelled by a binomial distribution.

The probability of a parking lot being occupied is $0.95$. Find the probability that out of the $80$ chosen parking lots, at least $90$% are occupied.

2013 MJC P2 Q8 [Modified]

<p>When a large number of mangoes are harvested, $37$% of them are unripe. The mangoes are packed randomly in boxes of $6$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the probability that there is no unripe mango in a box is $0.062524$, correct to $5$ significant figures.</div></li><li><div class="list-item-paragraph">Find the probability that there are exactly $2$ unripe mangoes in $2$ boxes.</div></li></ol>
<p>A supermarket orders $50$ boxes of mangoes daily.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the probability that there are at least $9$ boxes with no unripe mango in a particular day.</div></li><li><div class="list-item-paragraph">When the supermarket places a daily order for&nbsp;$10$ weeks, estimate the probability that the mean number of boxes with no unripe mango in a day is between $3$ and $9$.</div></li></ol>

When a large number of mangoes are harvested, $37$% of them are unripe. The mangoes are packed randomly in boxes of $6$.

Show that the probability that there is no unripe mango in a box is $0.062524$, correct to $5$ significant figures.

Find the probability that there are exactly $2$ unripe mangoes in $2$ boxes.

A supermarket orders $50$ boxes of mangoes daily.

Find the probability that there are at least $9$ boxes with no unripe mango in a particular day.

When the supermarket places a daily order for $10$ weeks, estimate the probability that the mean number of boxes with no unripe mango in a day is between $3$ and $9$.

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