A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>In a game, $3$ green balls and $7$ yellow balls are placed in a bag. A player draws $4$ balls at random and without replacement. The number of green balls that she draws is denoted by $G$. Find the probability distribution of $G$ and show that $P\left( G\ge 2 \right)=\frac{1}{3}$. Show that $E\left( G \right)=\frac{6}{5}$ and find the variance of $G$.<br></br></p>
<p>Alice scores $5$ points for each green ball that she draws and Bob scores $2$ points for each yellow ball that he draws.<br></br></p>
<p>Alice’s score is denoted by $A$ and Bob’s score is denoted by $B$.<br></br></p>
<p>Without finding the probability distribution of the number of yellow balls, find the expectation and variance of ${A-B}$  if</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Alice and Bob play two separate independent games to decide their scores,</div></li><li><div class="list-item-paragraph">Alice and Bob decide their score in the same game.</div></li></ol>

In a game, $3$ green balls and $7$ yellow balls are placed in a bag. A player draws $4$ balls at random and without replacement. The number of green balls that she draws is denoted by $G$. Find the probability distribution of $G$ and show that $P\left( G\ge 2 \right)=\frac{1}{3}$. Show that $E\left( G \right)=\frac{6}{5}$ and find the variance of $G$.<br></br>

Alice scores $5$ points for each green ball that she draws and Bob scores $2$ points for each yellow ball that he draws.<br></br>

Alice’s score is denoted by $A$ and Bob’s score is denoted by $B$.<br></br>

Without finding the probability distribution of the number of yellow balls, find the expectation and variance of ${A-B}$  if

Constructing and Using Discrete Probability Distributions

Expectation and Variance of Discrete Random Variables

Scaled Random Variables and Sums of Independent Copies

Linear Combinations of Random Variables

Discrete Random Variables (DRV)

<ol class="abc-list"><li><div class="list-item-paragraph">The discrete random variable $X$ takes values $-1$, $0$, $1$ with probabilities $\frac{1}{4}$, $\frac{1}{2}$, $\frac{1}{4}$ respectively. The variable $\bar{X}$ is the mean of a random sample of $3$ values of $X$ (i.e. ${{X}_{1}}$, ${{X}_{2}}$ and ${{X}_{3}}$ are independent random variables).</div><div class="list-item-paragraph">Tabulate the probability distribution of $\bar{X}$, and use your values to calculate $\mathrm{Var}\left( {\bar{X}} \right)$.</div><div class="list-item-paragraph">Hence verify that $\mathrm{Var}\left( {\bar{X}} \right)=\frac{1}{3}\mathrm{Var}\left( X \right)$ in this case.</div></li></ol>

Sample Mean and Its Sampling Distribution

N1984 P2 Q11

Sampling

<p>A computer randomly chooses $10$ shapes from $18$ squares and $12$ triangles. The number of squares chosen is denoted by $S$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760836044009-pb3skgh65" data-list-item-path="item-1760836044009-pb3skgh65"><div class="list-item-paragraph">The most probable number of squares that the computer chooses is denoted by $s$. By using the fact that&nbsp;$P\left( S=s \right)&gt;P\left( S=s+1 \right)$, show that&nbsp;$s$ satisfies the inequality</div><div class="list-item-paragraph" style="text-align: center">$\left( s+1 \right)\left( s+3 \right)&gt;\left( a-s \right)\left( b-s \right)$</div><div class="list-item-paragraph">where&nbsp;$a$ and $b$ are constants to be determined. Hence find the value of $s$.<span class="inline-marks">[4]</span></div></li></ol>
<p>In a game, the computer randomly chooses $6$ squares and $4$ triangles. The squares are numbered from $1$ to $6$ and the triangles are numbered from $1$ to $4$. A square and a triangle are randomly chosen. Let&nbsp;$X$ be the score which is defined as the difference (taken as always positive) between the numbers of the chosen square and triangle.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760836072225-u40wni23o" data-list-item-path="item-1760836072225-u40wni23o"><div class="list-item-paragraph">Find the probability distribution of $X$.<span class="inline-marks">[3]</span></div></li></ol>
<p>Three independent games are played. The total score of the games is $3$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-1760836085229-km6pguiff" data-list-item-path="item-1760836085229-km6pguiff"><div class="list-item-paragraph">Find the probability that the score of the first game is $2$.<span class="inline-marks">[3]</span></div></li></ol>

A computer randomly chooses $10$ shapes from $18$ squares and $12$ triangles. The number of squares chosen is denoted by $S$.

In a game, the computer randomly chooses $6$ squares and $4$ triangles. The squares are numbered from $1$ to $6$ and the triangles are numbered from $1$ to $4$. A square and a triangle are randomly chosen. Let $X$ be the score which is defined as the difference (taken as always positive) between the numbers of the chosen square and triangle.

Three independent games are played. The total score of the games is $3$.

Conditional Probability

Selections with Restrictions

Probability Using Permutations and Combinations

2024 NYJC P2 Q7

Probability

Permutation and Combination

<p>Four rods of lengths $1$, $2$, $3$ and $4$ units are placed in a bag from which one rod is selected at random. The probability of selecting a rod of length $l$ is $kl$. Find the value of $k$.</p>
<p>Show that the expected value of $X$, the length of the selected rod, is $3$ units and find the variance of $X$.</p>
<p>After a rod has been selected it is not replaced. The probabilities of selection for each of the three rods remain are in the same ratio as they were before the first selection. A second rod is now selected from the bag. Defining $Y$ to be the length of the rod and writing ${{P}_{1}}=\mathrm{P}\left( Y=1|X=2 \right)$, ${{P}_{2}}=\mathrm{P}\left( Y=2|X=1 \right)$ show that $16{{P}_{1}}=9{{P}_{2}}$.</p>
<p>Show also that $\mathrm{P}\left( X+Y=3 \right)=\frac{17}{360}$.</p>

Four rods of lengths $1$, $2$, $3$ and $4$ units are placed in a bag from which one rod is selected at random. The probability of selecting a rod of length $l$ is $kl$. Find the value of $k$.

Show that the expected value of $X$, the length of the selected rod, is $3$ units and find the variance of $X$.

After a rod has been selected it is not replaced. The probabilities of selection for each of the three rods remain are in the same ratio as they were before the first selection. A second rod is now selected from the bag. Defining $Y$ to be the length of the rod and writing ${{P}_{1}}=\mathrm{P}\left( Y=1|X=2 \right)$, ${{P}_{2}}=\mathrm{P}\left( Y=2|X=1 \right)$ show that $16{{P}_{1}}=9{{P}_{2}}$.

Show also that $\mathrm{P}\left( X+Y=3 \right)=\frac{17}{360}$.

Determining Unknowns in Discrete Probability Distributions

J2 Prelims Revision Set Problems Involving Lengths of Objects Q1

<p>Two coins are to be flipped. The first coin will land on heads with probability 0.6 and the second coin will land on heads with probability $0.7$. Assume that the results of the flips are independent. Every head obtained results in a prize, which is likely to be either a $\$10$ cash voucher or a $\$5$ cash voucher.</p>
<p>Let $X$ represent the total value of cash vouchers won.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $P\left( X=10 \right)=0.335$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Tabulate the probability distribution of $X$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the expectation of the total value of cash vouchers won.</div><p style="text-align: right;">[2]</p></li></ol>
<p>Let $D$ represent the number of $\$10$ cash vouchers won and $S$ represent the number of a $\$5$ cash vouchers. Use your answer in part <strong>(iii)</strong> to find $E\left( D \right)$and $E\left( S \right)$.</p><p style="text-align: right;">[2]</p>
<p>Explain why the same method cannot be used to find $\mathrm{Var}\left( D \right)$ and $\mathrm{Var}\left( S \right)$.</p><p style="text-align: right;">[1]</p>

Two coins are to be flipped. The first coin will land on heads with probability 0.6 and the second coin will land on heads with probability $0.7$. Assume that the results of the flips are independent. Every head obtained results in a prize, which is likely to be either a $\$10$ cash voucher or a $\$5$ cash voucher.

Let $X$ represent the total value of cash vouchers won.

Let $D$ represent the number of $\$10$ cash vouchers won and $S$ represent the number of a $\$5$ cash vouchers. Use your answer in part **(iii)** to find $E\left( D \right)$and $E\left( S \right)$.

Explain why the same method cannot be used to find $\mathrm{Var}\left( D \right)$ and $\mathrm{Var}\left( S \right)$.

2019 ACJC CT Q4

<p>At a particular game stall, players throw darts at a target board to earn points for redeeming prizes.</p>
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<p>The target board can be modelled by three concentric circles with radii one unit, three units and five units respectively. A player will get a score of $50$, $25$ or $0$ points if the dart lands on the respective regions labelled $50$, $25$ or $0$ as shown in the diagram above. The probability that a dart lands on any region labelled $50$, $25$ or $0$ is proportional to the area of the region. It can be assumed that the borders between the regions have negligible thickness and the dart will only land on the target board, which has a total area of $25\pi $ units$^{2}$.</p>
<p>In a game, a player throws two darts at the target board, one after the other. The first dart will be removed before the next dart is thrown. Let&nbsp;$S$ be the combined score that the player achieves from the two throws.</p>
<ol class="abc-list"><li data-list-item-id="item-1760448599889-kjjr3b7i1" data-list-item-path="item-1760448599889-kjjr3b7i1"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{P}\left( S=75 \right)=\frac{16}{625}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760448600480-dm3o33nu1" data-list-item-path="item-1760448600480-dm3o33nu1"><div class="list-item-paragraph">Find the probability distribution of $S$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760448601538-hyi94m572" data-list-item-path="item-1760448601538-hyi94m572"><div class="list-item-paragraph">Find the expectation and variance of $S$.<span class="inline-marks">[2]</span></div></li></ol>

At a particular game stall, players throw darts at a target board to earn points for redeeming prizes.

The target board can be modelled by three concentric circles with radii one unit, three units and five units respectively. A player will get a score of $50$, $25$ or $0$ points if the dart lands on the respective regions labelled $50$, $25$ or $0$ as shown in the diagram above. The probability that a dart lands on any region labelled $50$, $25$ or $0$ is proportional to the area of the region. It can be assumed that the borders between the regions have negligible thickness and the dart will only land on the target board, which has a total area of $25\pi $ units$^{2}$.

In a game, a player throws two darts at the target board, one after the other. The first dart will be removed before the next dart is thrown. Let $S$ be the combined score that the player achieves from the two throws.

2023 TMJC P2 Q6

<p>A biased $4$-sided die gives the scores $1$, $2$, $3$, and $4$ with the probabilities shown in the table, where&nbsp;$a$ and $b$ are constants. The random variable $S$ denotes the score on the die.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 500px;"><colgroup><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">Score $(s)$</th><th style="text-align: center; --cell-justify: center">$1$</th><th style="text-align: center; --cell-justify: center">$2$</th><th style="text-align: center; --cell-justify: center">$3$</th><th style="text-align: center; --cell-justify: center">$4$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$\mathrm{P}(S = s)$</td><td style="text-align: center; --cell-justify: center">$a$</td><td style="text-align: center; --cell-justify: center">$a$</td><td style="text-align: center; --cell-justify: center">$a$</td><td style="text-align: center; --cell-justify: center">$b$</td></tr></tbody></table></div></div>
<p>It is given that the mean of the score is $2.56$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;$b=\frac{7}{25}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">${{S}_{1}}$ and ${{S}_{2}}$ are two independent observations of $S$.</div><ol class="i-list"><li><div class="list-item-paragraph">Find the probability that the sum of the two independent observations of $S$ is at least $6$ given that one of the observed scores is $3$,<span class="inline-marks">[3]</span></div><div class="list-item-paragraph">Let&nbsp;$Y=\left| {{S}_{1}}-{{S}_{2}} \right|$. Without finding the probability distribution table of $Y$,</div></li><li><div class="list-item-paragraph">find $\mathrm{Var}\left( 2S-\mathrm{E}\left( Y \right) \right)$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">state the value of $\mathrm{E}\left( Y-\mathrm{E}\left( Y \right) \right)$.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

A biased $4$-sided die gives the scores $1$, $2$, $3$, and $4$ with the probabilities shown in the table, where $a$ and $b$ are constants. The random variable $S$ denotes the score on the die.

It is given that the mean of the score is $2.56$.

2023 HCI P2 Q8

<p>Tim plays a game with $3$ outcomes having the score of $2$, $5$ and $6$. The corresponding probabilities are $k$, $\frac{2}{7}$ and $\frac{4}{7}$ respectively, where&nbsp;$k$ is a constant.</p>
<ol class="abc-list"><li data-list-item-id="item-1760447775181-z41ubbpe8" data-list-item-path="item-1760447775181-z41ubbpe8"><div class="list-item-paragraph">State the value of $k$.<span class="inline-marks">[1]</span></div></li></ol>
<p>Tim pays $\$m$ in total to play the game twice. The outcome of each game is independent. He receives an amount corresponding to twice the absolute difference of the two scores from each game.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760447788306-gf3jjo83c" data-list-item-path="item-1760447788306-gf3jjo83c"><div class="list-item-paragraph">If Tim is expected make a profit, determine the range of values of $m$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760447792752-zr98pjsd3" data-list-item-path="item-1760447792752-zr98pjsd3"><div class="list-item-paragraph">For&nbsp;$m=2.5$, explain why Tim may not profit for every $2$ such games that he plays.<span class="inline-marks">[1]</span></div></li></ol>

Tim plays a game with $3$ outcomes having the score of $2$, $5$ and $6$. The corresponding probabilities are $k$, $\frac{2}{7}$ and $\frac{4}{7}$ respectively, where $k$ is a constant.

Tim pays $\$m$ in total to play the game twice. The outcome of each game is independent. He receives an amount corresponding to twice the absolute difference of the two scores from each game.

Expected Payoffs and Fair Games

2023 DHS P2 Q5

<ol class="i-list"><li><div class="list-item-paragraph">A multiple choice test consists of $20$ questions, each with four possible answers, of which only one is correct. If a student randomly chooses the answer to each question, find the probability of getting at least four but less than nine correct answers.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Suppose that each correct answer is awarded five marks and each incorrect answer carries a penalty of one mark, what is the expected score obtained by a student?</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Using a suitable approximation, find the probability that the mean score of $50$ randomly selected students is more than $12$ marks. State an assumption necessary for your answer to be valid.</div><p style="text-align: right;">[4]</p></li></ol>

Central Limit Theorem

Binomial Probabilities Using Technology

Assumptions for Sampling

Mean and Variance of a Binomial Distribution

NJC Sampling Tutorial Q4

<p>A computer can give independent observations of a random variable $X$ with probability distribution given by $\mathrm{P}\left( X=0 \right)=\frac{3}{4}$ and $\mathrm{P}\left( X=2 \right)=\frac{1}{4}$. It is programmed to output a value for the random variable $Y$ defined by $Y={{X}_{1}}+{{X}_{2}}$, where ${{X}_{1}}$ and ${{X}_{2}}$ are independent observations of $X$. Tabulate the probability distribution of $Y$, show that $E\left( Y \right)=1$.</p><p style="text-align: right;">[3]</p>
<p>The random variable $T$ is defined by $T={{Y}^{2}}$. Find $\mathrm{E}\left( T \right)$ and show that $\mathrm{Var}\left( T \right)=\frac{63}{4}$.</p><p style="text-align: right;">[3]</p>
<p>The computer is programmed to produce a large number $n$ of independent values of $T$ and to calculate the mean $M$ of these values. Find the smallest value of $n$ such that $\mathrm{P}\left( M<3 \right)>0.99$.</p><p style="text-align: right;">[4]</p>

A computer can give independent observations of a random variable $X$ with probability distribution given by $\mathrm{P}\left( X=0 \right)=\frac{3}{4}$ and $\mathrm{P}\left( X=2 \right)=\frac{1}{4}$. It is programmed to output a value for the random variable $Y$ defined by $Y={{X}_{1}}+{{X}_{2}}$, where ${{X}_{1}}$ and ${{X}_{2}}$ are independent observations of $X$. Tabulate the probability distribution of $Y$, show that $E\left( Y \right)=1$.

The random variable $T$ is defined by $T={{Y}^{2}}$. Find $\mathrm{E}\left( T \right)$ and show that $\mathrm{Var}\left( T \right)=\frac{63}{4}$.

The computer is programmed to produce a large number $n$ of independent values of $T$ and to calculate the mean $M$ of these values. Find the smallest value of $n$ such that $\mathrm{P}\left( M<3 \right)>0.99$.

Determining Sample Size from Probability Conditions

N1998 P2 Q7

<p>During an online mathematics test, multiple-choice questions are randomly selected from a large item bank. The probability that a selected question is on algebra is $\frac{4}{5}$ and that it is on geometry is $\frac{1}{5}$. Each question has four options, among which only one is correct. A candidate Amy, who has studied algebra but not geometry, answers each algebra question correctly with a probability of $\frac{7}{8}$. When faced with a geometry question, she simply makes a random guess. Four marks are awarded for each question answered correctly and one mark deducted for each question answered wrongly.</p>
<ol class="abc-list"><li data-list-item-id="item-1760342169030-mwds9xis5" data-list-item-path="item-1760342169030-mwds9xis5"><div class="list-item-paragraph">Consider a question attempted by Amy. Show that the probability that she answered the question correctly is $\frac{3}{4}$.</div></li><li data-list-item-id="item-1760342169720-21aoahzze" data-list-item-path="item-1760342169720-21aoahzze"><div class="list-item-paragraph">The random variable $X$ is the score obtained by Amy for a question. Write down the probability distribution of $X$.</div></li><li data-list-item-id="item-1760342170275-zvk5z4bjo" data-list-item-path="item-1760342170275-zvk5z4bjo"><div class="list-item-paragraph">Assume that Amy answers $50$ questions independently.</div><div class="list-item-paragraph">If Amy obtained at least $130$ marks, find the least number of questions that she answers correctly. Hence find the probability that Amy will obtain at least $130$ marks.</div></li></ol>

During an online mathematics test, multiple-choice questions are randomly selected from a large item bank. The probability that a selected question is on algebra is $\frac{4}{5}$ and that it is on geometry is $\frac{1}{5}$. Each question has four options, among which only one is correct. A candidate Amy, who has studied algebra but not geometry, answers each algebra question correctly with a probability of $\frac{7}{8}$. When faced with a geometry question, she simply makes a random guess. Four marks are awarded for each question answered correctly and one mark deducted for each question answered wrongly.

Law of Total Probability

Constructing and Using Probability Trees

2006 AJC P2 [Modified]

<p>Four rods of lengths $1$, $2$, $3$ and $4$ units are placed in a bag from which one rod is selected at random. The probability of selecting a rod of length $l$ is $kl$. Find the value of $k$.</p>
<p>Show that the expected value of $X$, the length of the selected rod, is $3$ units and the variance of $X$.</p>
<p>After a rod has been selected it is not replaced. The probabilities of selection for each of the three rods remain are in the same ratio as they were before the first selection. A second rod is now selected from the bag. Defining $Y$to be the length of the rod and writing ${{P}_{1}}=\mathrm{P}\left( Y=1|X=2 \right)$, ${{P}_{2}}=\mathrm{P}\left( Y=2|X=1 \right)$ show that $16{{P}_{1}}=9{{P}_{2}}$</p>
<p>Show also that $\mathrm{P}\left( X+Y=3 \right)=\frac{17}{360}$.</p>

Show that the expected value of $X$, the length of the selected rod, is $3$ units and the variance of $X$.

After a rod has been selected it is not replaced. The probabilities of selection for each of the three rods remain are in the same ratio as they were before the first selection. A second rod is now selected from the bag. Defining $Y$to be the length of the rod and writing ${{P}_{1}}=\mathrm{P}\left( Y=1|X=2 \right)$, ${{P}_{2}}=\mathrm{P}\left( Y=2|X=1 \right)$ show that $16{{P}_{1}}=9{{P}_{2}}$

Discrete Random Variable - Probability Involving Proportionality

<p>$X$ denotes the random variable representing the number of thunderstorms occurring per month. The probability distribution function of $X$ is given in the following table.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 350px;"><colgroup><col style="width: 100px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr><th style="text-align: left; --cell-justify: flex-start">$x$</th><th style="text-align: center; --cell-justify: center">$0$</th><th style="text-align: center; --cell-justify: center">$1$</th><th style="text-align: center; --cell-justify: center">$2$</th><th style="text-align: center; --cell-justify: center">$3$</th><th style="text-align: center; --cell-justify: center">$4$</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">$\mathrm{P}(X=x)$</td><td style="text-align: center; --cell-justify: center">$a$</td><td style="text-align: center; --cell-justify: center">$0.2$</td><td style="text-align: center; --cell-justify: center">$0.5$</td><td style="text-align: center; --cell-justify: center">$0.1$</td><td style="text-align: center; --cell-justify: center">$b$</td></tr></tbody></table></div></div>
<ol class="i-list"><li data-list-item-id="item-1760331550660-h4qhkl84c" data-list-item-path="item-1760331550660-h4qhkl84c"><div class="list-item-paragraph">State the values of $a$ and $b$ which maximizes the value of $\mathrm{E}\left( X \right)$.<span class="inline-marks">[2]</span></div></li></ol>
<p>Using these values of $a$ and $b$,</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1760331562735-4dc7ulmjp" data-list-item-path="item-1760331562735-4dc7ulmjp"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{Var}\left( X \right)=1.01$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760331564257-wjlaultfk" data-list-item-path="item-1760331564257-wjlaultfk"><div class="list-item-paragraph">Two independent observations ${{X}_{1}}$ and are taken of $X$. Find $\mathrm{P}\left( {{X}_{1}}-{{X}_{2}}\ge 1 \right)$.</div><div class="list-item-paragraph">Hence, or otherwise, find $\mathrm{P}\left( \left| {{X}_{1}}-{{X}_{2}} \right|\ge 1 \right)$.<span class="inline-marks">[5]</span></div></li></ol>

$X$ denotes the random variable representing the number of thunderstorms occurring per month. The probability distribution function of $X$ is given in the following table.

2005 RJC P2 Q25 [Modified]

<p>A fair die is thrown once and random variable $X$ denotes the score obtained. Write down $\mathrm{E}\left( X \right)$ and show that&nbsp;$\mathrm{Var}\left( X \right)=\frac{35}{12}$. The die is thrown twice, and ${{X}_{1}}$ and ${{X}_{2}}$ obtained from each attempt.</p>
<p>Tabulate the probability distribution of $Y=\left| {{X}_{1}}-{{X}_{2}} \right|$, and find $E\left( Y \right)$.</p>
<p>The random variable $Z$ is defined by $Z={{X}_{1}}-{{X}_{2}}$. State with reasons whether or not</p>
<ol class="i-list"><li><div class="list-item-paragraph">$E\left( {{Z}^{2}} \right)=E\left( {{Y}^{2}} \right)$.</div></li><li><div class="list-item-paragraph">$\mathrm{Var}\left( Z \right)=\mathrm{Var}\left( Y \right)$.</div></li></ol>

A fair die is thrown once and random variable $X$ denotes the score obtained. Write down $\mathrm{E}\left( X \right)$ and show that $\mathrm{Var}\left( X \right)=\frac{35}{12}$. The die is thrown twice, and ${{X}_{1}}$ and ${{X}_{2}}$ obtained from each attempt.

Tabulate the probability distribution of $Y=\left| {{X}_{1}}-{{X}_{2}} \right|$, and find $E\left( Y \right)$.

The random variable $Z$ is defined by $Z={{X}_{1}}-{{X}_{2}}$. State with reasons whether or not

N1978 P2 Q12

<p>A test consists of five multiple choice questions to each of which three answers are given, only one of which is correct. For each correct answer a candidate gets $2$ marks, but loses $1$ mark for each incorrect answer. A particular candidate answers all question purely by guesswork. Draw up a table to show all the possible total marks from $-5$ to $10$ and the probability associated with each mark for this candidate. Use the table to show that the expected mark is zero, and find the variance of this distribution.</p>

A test consists of five multiple choice questions to each of which three answers are given, only one of which is correct. For each correct answer a candidate gets $2$ marks, but loses $1$ mark for each incorrect answer. A particular candidate answers all question purely by guesswork. Draw up a table to show all the possible total marks from $-5$ to $10$ and the probability associated with each mark for this candidate. Use the table to show that the expected mark is zero, and find the variance of this distribution.

Binomial Probabilities Using the Probability Mass Function

N1977 P1 Q12

<p>A fair six-sided die and another fair tetrahedral (4 sided) die are thrown together once. $S$ is the absolute difference of the scores of the two dice.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the probability distribution of $S$.</div><p style="text-align: right;">[3]</p></li></ol>
<p>If ${{S}_{1}}$, ${{S}_{2}}$, ${{S}_{3}}$ are independent observations of $S$, find</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">$\mathrm{P}\left( {{S}_{1}}=\left. 2 \right|{{S}_{1}}+{{S}_{2}}+{{S}_{3}}=3 \right)$.</div><p style="text-align: right;">[3]</p></li></ol>

A fair six-sided die and another fair tetrahedral (4 sided) die are thrown together once. $S$ is the absolute difference of the scores of the two dice.

If ${{S}_{1}}$, ${{S}_{2}}$, ${{S}_{3}}$ are independent observations of $S$, find

2022 ACJC J2 MYE P1 Q11

<p>A bag contains nine numbered discs. Three discs are numbered $3$, four discs are numbered $4$ and two discs are numbered $-1$. Two discs are drawn simultaneously. The sum of numbers on them, denoted by $X$, is recorded.</p>
<ol class="i-list"><li data-list-item-id="item-1760165330641-llooic27f" data-list-item-path="item-1760165330641-llooic27f"><div class="list-item-paragraph">Find the probability distribution for&nbsp;$X$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760165331143-co3vknicw" data-list-item-path="item-1760165331143-co3vknicw"><div class="list-item-paragraph">Find $\mathrm{E}\left( X \right)$ and $\mathrm{Var}\left( X \right)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760165332148-pl4cpymmi" data-list-item-path="item-1760165332148-pl4cpymmi"><div class="list-item-paragraph">Two independent observations of $X$ are taken. Find the probability that the difference between these two values is at most $5$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760165333116-msley4a1x" data-list-item-path="item-1760165333116-msley4a1x"><div class="list-item-paragraph">Fifty independent observations of $X$ are taken. Find the approximate probability that the sum of these fifty observations is between $250$ and $260$.<span class="inline-marks">[3]</span></div></li></ol>

A bag contains nine numbered discs. Three discs are numbered $3$, four discs are numbered $4$ and two discs are numbered $-1$. Two discs are drawn simultaneously. The sum of numbers on them, denoted by $X$, is recorded.

2022 ASRJC P2 Q10

<p>A computer chip generates a random number from the set $\left\{ 1,2,3,4 \right\}$. The probability distribution of the number generated, $X$, is given by</p>
<p style="text-align: center">$\mathrm{P}\left( X=x \right)=\left\{ \begin{matrix}
p\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x=1 \\
\frac{1-p}{3}\,\,\,\,\,\,\,\,\,\,x=2,3,4 \\
\end{matrix} \right.$</p>
<p>where $0 < p < 1$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find $\mathrm{E}\left( X \right)$ in terms of $p$ and hence show that $\mathrm{Var}\left( X \right)=\frac{2}{3}\left( 1-p \right)\left( 1+6p \right)$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">${{X}_{1}}$ and ${{X}_{2}}$ are two independent observations of $X$. Find $\mathrm{P}\left( \left| {{X}_{1}}-{{X}_{2}} \right|>1 \right)$ in terms of $p$. Hence, find the maximum possible value of this probability as $p$ varies.</div><p style="text-align: right;">[5]</p></li></ol>
<p>It is given instead that $p=\frac{1}{2}$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the probability that the sum of $50$ independent observations of $X$ exceeds $110$.</div><p style="text-align: right;">[3]</p></li></ol>

A computer chip generates a random number from the set $\left\{ 1,2,3,4 \right\}$. The probability distribution of the number generated, $X$, is given by

$\mathrm{P}\left( X=x \right)=\left\{ \begin{matrix}
p\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x=1 \\
\frac{1-p}{3}\,\,\,\,\,\,\,\,\,\,x=2,3,4 \\
\end{matrix} \right.$

It is given instead that $p=\frac{1}{2}$.

2020 EJC P2 Q9

<p>A biscuit manufacturer produces both cream and chocolate biscuits. Biscuits are chosen randomly and packed into boxes of $10$. The number of cream biscuits in a box is denoted by $X$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">On average, the proportion of cream biscuits is $p$. Given that $\mathrm{P}\left( X=1\,\,\mathrm{or}\,\,2 \right)=0.15$, write down an equation for the value of $p$. Hence find the value(s) of $p$ numerically.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">It is given instead that the biscuit manufacturer produces $3$ times as many cream biscuits as chocolate biscuits.</div><ol class="i-list"><li><div class="list-item-paragraph">Find the most likely value of $X$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">A random sample of $18$ boxes is taken. Find the probability that at least $3$ but fewer than $7$ boxes have equal numbers of cream and chocolate biscuits.</div><p style="text-align: right;">[3]</p></li></ol></li></ol>
<p>A box of biscuits is sold at $\$10$. The manufacturer gives a discount of $\$2$ per box to its premium customers. The mean and variance of the number of boxes sold per day to each type of customers (assuming independence) are as follows:</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759832233144-khay4yq75mo.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTk4MzIyMzMxNDQta2hheTR5cTc1bW8ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUyODAwNCwiZXhwIjoxNzkzNTI4MzA0fQ.0b8rNdRy_Xj4ytt9fVPsGhyRjATA4yKQQ11sL_uHZrA" width="400" class="aligncenter" /></figure>
<p>Find the approximate probability that the total amount collected per month from the sales of biscuits is not less than $\$255,000$, assuming that there are $30$ days in a month.</p><p style="text-align: right;">[4]</p>

A biscuit manufacturer produces both cream and chocolate biscuits. Biscuits are chosen randomly and packed into boxes of $10$. The number of cream biscuits in a box is denoted by $X$.

A box of biscuits is sold at $\$10$. The manufacturer gives a discount of $\$2$ per box to its premium customers. The mean and variance of the number of boxes sold per day to each type of customers (assuming independence) are as follows:

Find the approximate probability that the total amount collected per month from the sales of biscuits is not less than $\$255,000$, assuming that there are $30$ days in a month.

Mode of a Binomial Distribution

Repeated Binomial Experiments

Determining an Unknown Binomial Success Probability

2017 AJC P2 Q8

<p>The probability of obtaining a ‘$6$’ on a biased cubical dice is thrice the probability of rolling every other number on the dice.</p>
<p>Show that the probability of obtaining a ‘$6$’ on the biased dice is $\frac{3}{8}$.<span class="inline-marks">[1]</span></p>
<p>This biased dice is put into a bag together with $3$ fair cubical dice.</p>
<ol class="i-list"><li data-list-item-id="item-1759814610426-yojbikwur" data-list-item-path="item-1759814610426-yojbikwur"><div class="list-item-paragraph">One of the dice is chosen randomly from the bag and rolled once. Find the probability of obtaining a ‘$6$’.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759814611003-c52ba1y7e" data-list-item-path="item-1759814611003-c52ba1y7e"><div class="list-item-paragraph">One of the dice is chosen randomly and is rolled $n$ times.</div><ol class="abc-list"><li data-list-item-id="subitem-1759814614506-ey4h222i0" data-list-item-path="item-1759814611003-c52ba1y7e/subitem-1759814614506-ey4h222i0"><div class="list-item-paragraph">Find the probability that ‘$6$’ is obtained on all the $n$ rolls.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1759814615564-939f0rvnc" data-list-item-path="item-1759814611003-c52ba1y7e/subitem-1759814615564-939f0rvnc"><div class="list-item-paragraph">Given that ‘$6$’ is obtained for all the $n$ rolls, the probability that the biased dice is chosen is more than $0.95$. By forming an inequality in terms of $n$, solve for the least value of $n$.<span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-1759814611933-fqehhktyk" data-list-item-path="item-1759814611933-fqehhktyk"><div class="list-item-paragraph">One of the dice is chosen randomly from the bag and rolled once. The dice is then put back into the bag. Another dice is chosen randomly from the bag and rolled once. For every ‘$6$’ obtained, the player gains $\$2$, otherwise the player loses</div><div class="list-item-paragraph">$\$k$. If the game is fair, that is, expected winnings of the player is $\$0$, show that the value of $k$ is $0.56$.<span class="inline-marks">[3]</span></div><div class="list-item-paragraph">Find the variance of the player’s winnings.<span class="inline-marks">[2]</span></div></li></ol>

The probability of obtaining a ‘$6$’ on a biased cubical dice is thrice the probability of rolling every other number on the dice.

Show that the probability of obtaining a ‘$6$’ on the biased dice is $\frac{3}{8}$.

This biased dice is put into a bag together with $3$ fair cubical dice.

Math Question Bank