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>The probability distribution of a discrete random variable $X$ is given by</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 390px;"><colgroup><col style="width: 38.4615%;"><col style="width: 17.9487%;"><col style="width: 17.9487%;"><col style="width: 25.6410%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">$x$</th><th style="text-align: center; --cell-justify: center">$0$</th><th style="text-align: center; --cell-justify: center">$2$</th><th style="text-align: center; --cell-justify: center">$4$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$P(X=x)$</td><td style="text-align: center; --cell-justify: center">$p$</td><td style="text-align: center; --cell-justify: center">$p$</td><td style="text-align: center; --cell-justify: center">$1-2p$</td></tr></tbody></table></div></div>
<ol class="i-list"><li data-list-item-id="vm-q2-i" data-list-item-path="vm-q2-i"><div class="list-item-paragraph">State the range of possible values of $p$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="vm-q2-ii" data-list-item-path="vm-q2-ii"><div class="list-item-paragraph">Find the maximum and minimum possible values of $E(X)$, stating the corresponding value of $p$ in each case.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="vm-q2-iii" data-list-item-path="vm-q2-iii"><div class="list-item-paragraph">Show that&nbsp;$\operatorname{Var}(X)=4p(5-9p)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="vm-q2-iv" data-list-item-path="vm-q2-iv"><div class="list-item-paragraph">Hence find the maximum and minimum possible values of $\operatorname{Var}(X)$, stating all corresponding values of $p$.<span class="inline-marks">[3]</span></div></li></ol>

The probability distribution of a discrete random variable $X$ is given by

Constructing and Using Discrete Probability Distributions

Expectation and Variance of Discrete Random Variables

Determining Unknowns in Discrete Probability Distributions

Discrete Random Variables (DRV)

<p>A discrete random variable $X$ takes only the values $-3$ and $5$. It is given that&nbsp;$P(X=5)=p$ and $P(X=-3)=1-p$, where&nbsp;$0\le p\le 1$.</p>
<ol class="i-list"><li data-list-item-id="vm-q1-i" data-list-item-path="vm-q1-i"><div class="list-item-paragraph">Show that&nbsp;$E(X)=-3+8p$ and $\operatorname{Var}(X)=64p(1-p)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="vm-q1-ii" data-list-item-path="vm-q1-ii"><div class="list-item-paragraph">Find the maximum and minimum possible values of $E(X)$, stating the corresponding value of $p$ in each case.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="vm-q1-iii" data-list-item-path="vm-q1-iii"><div class="list-item-paragraph">Find the maximum and minimum possible values of $\operatorname{Var}(X)$, stating all corresponding values of $p$.<span class="inline-marks">[3]</span></div></li></ol>

A discrete random variable $X$ takes only the values $-3$ and $5$. It is given that $P(X=5)=p$ and $P(X=-3)=1-p$, where $0\le p\le 1$.

<p>The probability density function of the discrete random variable $X$ is given by</p>
<p>\[P(X=x)=a\left(\frac{3}{4}\right)^x \text{ for}\hspace{0.5em} x=0,1,2,...\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the value of the constant $a$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the values of $P(X&lt;3)$ and $P(X&gt;2)$.</span></div></li></ol>

The probability density function of the discrete random variable $X$ is given by

\[P(X=x)=a\left(\frac{3}{4}\right)^x \text{ for } x=0,1,2,...\]

Find the values of $P(X<3)$ and $P(X>2)$.

Finding unknown constant in a probability distribution function

<p class="paragraph-header">IB AA SL: core and optional content</p>
<p>Core: normalization and expectation. Optional: variance.</p>
<p>A random variable $X$ takes integer values between $1$ and $4$ inclusive where&nbsp;$\text{P}\hspace{0.1em}\left( X=x \right)=a{{x}^{2}}$ for&nbsp;$x=1,\text{}\hspace{0.5em}2,\text{}\hspace{0.5em}3,\text{}\hspace{0.5em}4$. <br>Find $\text{E}\hspace{0.1em}\left( X \right)$ and $\text{Var}\hspace{0.1em}\left( X \right)$.</p>

Core: normalization and expectation. Optional: variance.

A random variable $X$ takes integer values between $1$ and $4$ inclusive where $\text{P}\left( X=x \right)=a{{x}^{2}}$ for $x=1,\text{ }2,\text{ }3,\text{ }4$. 
Find $\text{E}\left( X \right)$ and $\text{Var}\left( X \right)$.

<p class="paragraph-header">IB AA SL: core and optional content</p>
<p>Core: normalization and expectation. Optional: variance.</p>
<p>A random variable $X$ takes integer values between $1$ and $4$ inclusive where&nbsp;$\mathrm{P}\left( X=x \right)=a x^{3}+0.01$ for&nbsp;$x=1,\ 2,\ 3,\ 4$.</p>
<p>Find $\mathrm{E}\left( X \right)$ and $\mathrm{Var}\left( X \right)$.</p>

A random variable $X$ takes integer values between $1$ and $4$ inclusive where $\mathrm{P}\left( X=x \right)=a x^{3}+0.01$ for $x=1,\ 2,\ 3,\ 4$.

Find $\mathrm{E}\left( X \right)$ and $\mathrm{Var}\left( X \right)$.

<p>A box contains two balls numbered $2$ and $k$ balls numbered $3$, where&nbsp;$k\ge 3$. In a game, three balls are drawn from the box at random and the score is obtained by multiplying together the numbers indicated on the balls that were drawn. Let the random variable $X$ denote the score obtained.</p>
<ol class="abc-list"><li data-list-item-id="item-1760847139471-5n3754fq8" data-list-item-path="item-1760847139471-5n3754fq8"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{E}\left( X \right)=\frac{9\left( 3{{k}^{2}}+3k+2 \right)}{\left( k+1 \right)\left( k+2 \right)}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760847140266-tshe7xjxf" data-list-item-path="item-1760847140266-tshe7xjxf"><div class="list-item-paragraph">Given that&nbsp;$\mathrm{E}\left( X \right)=\frac{144}{7}$, find the value of $k$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760847141833-1l2lm15gf" data-list-item-path="item-1760847141833-1l2lm15gf"><div class="list-item-paragraph">Assuming that each game is independent of one another, find the probability that the average score of $30$ games is more than $21$.<span class="inline-marks">[3]</span></div></li></ol>

A box contains two balls numbered $2$ and $k$ balls numbered $3$, where $k\ge 3$. In a game, three balls are drawn from the box at random and the score is obtained by multiplying together the numbers indicated on the balls that were drawn. Let the random variable $X$ denote the score obtained.

Central Limit Theorem

Probability Using Permutations and Combinations

Sample Mean and Its Sampling Distribution

2024 YIJC P2 Q9

Probability

Sampling

<p>At a carnival, a game stall offers a game of throwing darts. The darts will land either in the bull’s eye, inner ring or outer ring. There is a probability of $p$, $2p$ and $5p$ of landing in the bull’s eye, inner ring and outer ring respectively.</p>
<p>A player gets five points if the dart lands in the bull’s eye, three points if the dart lands in the inner ring and one point if the dart lands in the outer ring. A round consists of two throws. Let&nbsp;$X$ be the total number of points for one round of two throws.</p>
<ol class="i-list"><li data-list-item-id="item-1760836156284-hz3nk1203" data-list-item-path="item-1760836156284-hz3nk1203"><div class="list-item-paragraph">Find the value of $p$ and show that&nbsp;$P\left( X=6 \right)=\frac{7}{32}$. Obtain the probability distribution of $X$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-1760836157912-6sl5k5818" data-list-item-path="item-1760836157912-6sl5k5818"><div class="list-item-paragraph">Find $\mathrm{E}\left( X \right)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760836158542-9n4g7izwt" data-list-item-path="item-1760836158542-9n4g7izwt"><div class="list-item-paragraph">A player wins $\$1$ for every two points earned. If a player pays $\$k$ to play one round of two throws, find the value of $k$ for the game to be fair.<span class="inline-marks">[2]</span></div></li></ol>

At a carnival, a game stall offers a game of throwing darts. The darts will land either in the bull’s eye, inner ring or outer ring. There is a probability of $p$, $2p$ and $5p$ of landing in the bull’s eye, inner ring and outer ring respectively.

A player gets five points if the dart lands in the bull’s eye, three points if the dart lands in the inner ring and one point if the dart lands in the outer ring. A round consists of two throws. Let $X$ be the total number of points for one round of two throws.

Expected Payoffs and Fair Games

2024 TMJC P2 Q6

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

Conditional Probability

Linear Combinations of Random Variables

J2 Prelims Revision Set Problems Involving Lengths of Objects Q1

<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.

2023 DHS P2 Q5

<p>Antony plays a carnival game. In the game, there is a box containing $3$ tokens numbered $4$ and $n$ tokens numbered $1$, where&nbsp;$n\ge 3$. Antony takes three tokens at random and without replacement. His score is the sum of the numbers on the three tokens taken.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Determine the probability distribution of Antony’s score.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Given that the expectation of Antony’s score is $4.5$, find the value of $n$.<span class="inline-marks">[3]</span></div></li></ol>

Antony plays a carnival game. In the game, there is a box containing $3$ tokens numbered $4$ and $n$ tokens numbered $1$, where $n\ge 3$. Antony takes three tokens at random and without replacement. His score is the sum of the numbers on the three tokens taken.

2022 RVHS P2 Q7

<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>Andrew plays a game with 4 cards numbered 1, 2 , 3 and 4. In each round, a random sample of $n$ cards is taken without replacement, where $1\le n\le 4$ and there is equal chance of Andrew taking 1, 2, 3 or 4 cards. Let $L$ denote the smallest value of the chosen cards. The distribution table for $L$ is as follows:</p>
<table><thead><tr><th>$L$</th><th>$1$</th><th>$2$</th><th>$3$</th><th>$4$</th></tr></thead><tbody><tr><td>${\mathbb{P}(L = l)}$</td><td>$a$</td><td>$b$</td><td>${\frac{5}{48}}$</td><td>${\frac{1}{16}}$</td></tr></tbody></table>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $b=\frac{5}{24}$.</div></li><li><div class="list-item-paragraph">Show that $\mathrm{Var}\left( L \right)=\frac{1895}{2304}$.</div></li><li><div class="list-item-paragraph">Andrew invites Jasmine to join in the game. In each round, only 1 player plays and the cards are replaced before the other player plays the game. Jasmine has the first move in the game. The first player to obtain the results $L=1$ will win the game. Find the probability that Jasmine wins the game.</div></li></ol>

Andrew plays a game with 4 cards numbered 1, 2 , 3 and 4. In each round, a random sample of $n$ cards is taken without replacement, where $1\le n\le 4$ and there is equal chance of Andrew taking 1, 2, 3 or 4 cards. Let $L$ denote the smallest value of the chosen cards. The distribution table for $L$ is as follows:

Repeated Trials with Infinitely Many Possible Outcomes

Law of Total Probability

Waiting-Time and Stopping-Rule Probabilities

2006 HCI P2 Q28 [Modified]

<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>Emma has a computer program that generates a random positive integer $X$. The probability distribution of $X$ is:</p>
<p style="text-align: center">$\mathrm{P}\left( X=r \right)=\frac{a}{{{r}^{3}}}$, $r\in {{\mathbb{Z}}^{+}}$ and $a$ is positive constant.</p>
<p>For the rest of the question, you may use the following results:</p>
<p style="text-align: center">$\sum\limits_{m=1}^{\infty }{\frac{1}{m}}$ does not exist, $\sum\limits_{m=1}^{\infty }{\frac{1}{{{m}^{2}}}=\mathrm{1}\mathrm{.6449}}$ and $\sum\limits_{m=1}^{\infty }{\frac{1}{{{m}^{3}}}=\mathrm{1}\mathrm{.2021}}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the value of $a$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find $\mathrm{E}\left( X \right)$ and explain why $\mathrm{Var}\left( X \right)$ cannot be calculated.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find $\mathrm{P}\left( \left. X\ge 2 \right|X\le 15 \right)$.</div><p style="text-align: right;">[3]</p></li></ol>
<p>Emma generates $10$ numbers using her program namely, ${{X}_{1}}$, ${{X}_{2}}$, ... ,${{X}_{10}}$. The random variable $Y$ denotes the number of times the number ‘$3$’ occurs among the $10$ numbers. You can assume that the numbers generated by the program are independent of each other.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find $\mathrm{P}\left( Y>2 \right)$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find $\mathrm{P}\left( {{X}_{1}}+Y=3 \right)$.</div><p style="text-align: right;">[4]</p></li></ol>

Emma has a computer program that generates a random positive integer $X$. The probability distribution of $X$ is:

$\mathrm{P}\left( X=r \right)=\frac{a}{{{r}^{3}}}$, $r\in {{\mathbb{Z}}^{+}}$ and $a$ is positive constant.

For the rest of the question, you may use the following results:

$\sum\limits_{m=1}^{\infty }{\frac{1}{m}}$ does not exist, $\sum\limits_{m=1}^{\infty }{\frac{1}{{{m}^{2}}}=\mathrm{1}\mathrm{.6449}}$ and $\sum\limits_{m=1}^{\infty }{\frac{1}{{{m}^{3}}}=\mathrm{1}\mathrm{.2021}}$.

Emma generates $10$ numbers using her program namely, ${{X}_{1}}$, ${{X}_{2}}$, ... ,${{X}_{10}}$. The random variable $Y$ denotes the number of times the number ‘$3$’ occurs among the $10$ numbers. You can assume that the numbers generated by the program are independent of each other.

Binomial Probabilities Using Technology

2022 DHS P2 Q11

<p>A board of directors consists $n$ men and $\left( 10-n \right)$ women, where&nbsp;$3\le n\le 7$. A committee consisting $3$ randomly chosen people is to be formed. Let&nbsp;$W$ be the number of women in the committee.</p>
<ol class="i-list"><li data-list-item-id="item-1760152845049-o4qyl2ayp" data-list-item-path="item-1760152845049-o4qyl2ayp"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{P}\left( W=1 \right)=\frac{n\left( n-1 \right)\left( 10-n \right)}{240}$ and find, in terms of $n$, the probability distribution of $W$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760152845634-rp282xf5l" data-list-item-path="item-1760152845634-rp282xf5l"><div class="list-item-paragraph">Given&nbsp;$E\left( W \right)=2.1$, find $n$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760152846392-kj18wpkxr" data-list-item-path="item-1760152846392-kj18wpkxr"><div class="list-item-paragraph">Hence, find $\mathrm{Var}\left( W \right)$.<span class="inline-marks">[1]</span></div></li></ol>

A board of directors consists $n$ men and $\left( 10-n \right)$ women, where $3\le n\le 7$. A committee consisting $3$ randomly chosen people is to be formed. Let $W$ be the number of women in the committee.

Selections with Restrictions

2018 VJC J2 MYE P1 Q10

Permutation and Combination

<p>The discrete random variable $X$ has probability mass function given by</p>
<p style="text-align: center">$\mathrm{P}(X=x)=\left\{ \begin{matrix} & \frac{ax}{n(n+1)}\,\,\,\,\mathrm{for}\,\,x=1,2,...,n, \\ & 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{otherwise,} \\ \end{matrix} \right.$</p>
<p>where $a$ is a constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $a=2$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find $\mathrm{E}(X)$ in terms of $n$. You may use the result that $\sum\limits_{r=1}^{n}{{{r}^{2}}=\frac{n}{6}(n+1)(2n+1)}$.</div><p style="text-align: right;">[3]</p></li></ol>
<p>Two players ${{P}_{1}}$ and ${{P}_{2}}$ play a game with $n+1$ tokens labelled $1$ to $n+1$. Each player randomly picks one token without replacement and the player who picks the token with the smaller number loses. The amount of money lost by the losing player, in dollars, will be the number on the winning token. For example, if ${{P}_{1}}$ and ${{P}_{2}}$ pick the tokens labelled $5$ and $3$ respectively, ${{P}_{2}}$ loses $\$\,5$ to ${{P}_{1}}$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Explain why the probability of ${{P}_{2}}$ losing a game is $0.5$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Given that ${{P}_{2}}$ loses, find the probability ${{P}_{2}}$ loses $\$\,(m+1)$ in terms of $m$ and $n$, where $m$ is such that $1\le m\le n$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Using the result in part (i), when ${{P}_{2}}$ loses a game, find the amount that he is expected to lose in terms of $n$.</div><p style="text-align: right;">[3]</p></li></ol>

The discrete random variable $X$ has probability mass function given by

$\mathrm{P}(X=x)=\left\{ \begin{matrix} & \frac{ax}{n(n+1)}\,\,\,\,\mathrm{for}\,\,x=1,2,...,n, \\ & 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{otherwise,} \\ \end{matrix} \right.$

Two players ${{P}_{1}}$ and ${{P}_{2}}$ play a game with $n+1$ tokens labelled $1$ to $n+1$. Each player randomly picks one token without replacement and the player who picks the token with the smaller number loses. The amount of money lost by the losing player, in dollars, will be the number on the winning token. For example, if ${{P}_{1}}$ and ${{P}_{2}}$ pick the tokens labelled $5$ and $3$ respectively, ${{P}_{2}}$ loses $\$\,5$ to ${{P}_{1}}$.

2018 DHS P2 Q10

<p>A bag contains three yellow marbles, one blue marble and $x$ red marbles, where&nbsp;$x&gt;1$. In a game, Lily takes $~2$ marbles at random from the bag, without replacement. She scores $~2$ points for each red marble taken and $1$ point for each yellow marble taken. If a blue marble is taken, she loses $~2$ points. The random variable $S$ is the sum of the points of the two marbles taken.</p>
<ol class="i-list"><li data-list-item-id="item-1760093524192-y7eti6eek" data-list-item-path="item-1760093524192-y7eti6eek"><div class="list-item-paragraph">Find $\mathrm{P}\left( S=s \right)$ for all possible values of $s$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760093525152-ms43glzvs" data-list-item-path="item-1760093525152-ms43glzvs"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{E}\left( S \right)=\frac{2\left( 2x+1 \right)}{x+4}$ and find an expression for&nbsp;$\mathrm{Var}\left( S \right)$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-1760093526127-aby0wk5js" data-list-item-path="item-1760093526127-aby0wk5js"><div class="list-item-paragraph">Find the least value of $x$ if&nbsp;$\mathrm{E}\left( S \right)&gt;2$.<span class="inline-marks">[2]</span></div></li></ol>

A bag contains three yellow marbles, one blue marble and $x$ red marbles, where $x>1$. In a game, Lily takes $~2$ marbles at random from the bag, without replacement. She scores $~2$ points for each red marble taken and $1$ point for each yellow marble taken. If a blue marble is taken, she loses $~2$ points. The random variable $S$ is the sum of the points of the two marbles taken.

2022 NYJC P2 Q9

<p>random variable $X$ has probability distribution given by</p>
<p style="text-align: center">$\mathrm{P}\left( X=x \right)=\left\{ \begin{matrix}
kx\left( 6-x \right),\,\,\,x=1,2,3,4,5 \\
0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{otherwise}\mathrm{.} \\
\end{matrix} \right.$</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $k=\frac{1}{35}$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the mean and variance of $X$.</div><p style="text-align: right;">[3]</p></li></ol>
<p>$40$ independent observations of $X$ are taken.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the probability that the sum of the $40$ observations of $X$ exceeds $110$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the probability that the maximum value observed is $5$.</div><p style="text-align: right;">[3]</p></li></ol>

random variable $X$ has probability distribution given by

$\mathrm{P}\left( X=x \right)=\left\{ \begin{matrix}
kx\left( 6-x \right),\,\,\,x=1,2,3,4,5 \\
0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{otherwise}\mathrm{.} \\
\end{matrix} \right.$

$40$ independent observations of $X$ are taken.

2021 TJC P2 Q9

<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.

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