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>A committee of $10$ people is chosen at random from a group consisting of $18$ women and $12$ men. The number of women on the committee is denoted by $R$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that&nbsp;$R = 4$.</span></div></li><li><div class="list-item-paragraph">The most probable number of women on the committee is denoted by $r$. By using the fact that&nbsp;$ \mathrm{P}(R = r) &gt; \mathrm{P}(R = r + 1)$, show that&nbsp;$r$ satisfies the inequality \[(r + 1)! (17 - r)! (9 - r)! (r + 3)! &gt; r! (18 - r)! (10 - r)! (r + 2)!\] and use this inequality to find the value of $r$.</div></li><li><div class="list-item-paragraph">Use the calculator to find $\mathrm{E}(R)$ and $\mathrm{Var}(R)$.</div></li></ol>

A committee of $10$ people is chosen at random from a group consisting of $18$ women and $12$ men. The number of women on the committee is denoted by $R$.

Selections with Restrictions

Probability Using Permutations and Combinations

N2011 P2 Q11 [Modified]

Probability

Permutation and Combination

<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

Scaled Random Variables and Sums of Independent Copies

Linear Combinations of Random Variables

<p>Prove that&nbsp;$\mathrm{Var}\left( X \right)=\mathrm{E}\left( X^{2} \right)-\left[ \mathrm{E}\left( X \right) \right]^{2}$.</p>

Prove that $\mathrm{Var}\left( X \right)=\mathrm{E}\left( X^{2} \right)-\left[ \mathrm{E}\left( X \right) \right]^{2}$.

<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 class="paragraph-header">IB AA SL: core and optional content</p>
<p>Core: distribution table and expectation. Optional: variance.</p>
<p>There are $8$ red balls, $6$ black balls and $2$ blue balls in a bag. $2$ balls are drawn from the bag <strong>without </strong>replacement.</p>
<p>Let&nbsp;$X$ be the random variable of number of black balls drawn. Construct the probability distribution table for&nbsp;$X$ and show that&nbsp;$X$ is a discrete random variable. Find $\mathrm{E}\left( X \right)$ and $\mathrm{Var}\left( X \right)$.</p>

Core: distribution table and expectation. Optional: variance.

There are $8$ red balls, $6$ black balls and $2$ blue balls in a bag. $2$ balls are drawn from the bag **without **replacement.

Let $X$ be the random variable of number of black balls drawn. Construct the probability distribution table for $X$ and show that $X$ is a discrete random variable. Find $\mathrm{E}\left( X \right)$ and $\mathrm{Var}\left( X \right)$.

Discrete and Continuous Random Variables

<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 $\mathrm{E}\left( G \right)=\frac{6}{5}$ and find the variance of $G$.</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.</p>
<p>Alice’s score is denoted by $A$ and Bob’s score is denoted by $B$.</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 $\mathrm{E}\left( G \right)=\frac{6}{5}$ and find the variance of $G$.

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

Alice’s score is denoted by $A$ and Bob’s score is denoted by $B$.

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

Worked Example 4

<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 $x=1,\mathrm{ }2,\mathrm{ }3,\mathrm{ }4$.</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 $x=1,\mathrm{ }2,\mathrm{ }3,\mathrm{ }4$.

Worked Example 2

<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

Sample Mean and Its Sampling Distribution

2024 YIJC P2 Q9

Sampling

<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>

N1984 P2 Q11

<p>At a funfair game stall, players are allowed to choose two cards at random from six cards, with each card labelled with one letter from A to F. The player’s score, denoted by $X$, is the Manhattan distance between the two squares corresponding to the player’s two chosen letters on the grid below,</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 135px;"><colgroup><col style="width: 45px;"><col style="width: 45px;"><col style="width: 45px;"></colgroup><thead><tr style="height: 45px;"><th style="height: 45px; min-height: 45px; text-align: center; --cell-justify: center">$\mathrm{A}$</th><th style="height: 45px; min-height: 45px; text-align: center; --cell-justify: center">$\mathrm{B}$</th><th style="height: 45px; min-height: 45px; text-align: center; --cell-justify: center">$\mathrm{C}$</th></tr></thead><tbody><tr style="height: 45px;"><td style="height: 45px; min-height: 45px; text-align: center; --cell-justify: center">$\mathrm{D}$</td><td style="height: 45px; min-height: 45px; text-align: center; --cell-justify: center">$\mathrm{E}$</td><td style="height: 45px; min-height: 45px; text-align: center; --cell-justify: center">$\mathrm{F}$</td></tr></tbody></table></div></div>
<p>where the Manhattan distance between two squares is the minimum total number of horizontal and vertical steps required to travel between them. For example, the Manhattan distance between B and F is $2$, while the Manhattan distance between D and C is $3$.</p>
<ol class="i-list"><li data-list-item-id="item-1760836232049-ow62f7kkh" data-list-item-path="item-1760836232049-ow62f7kkh"><div class="list-item-paragraph">Tabulate the probability distribution of $X$.<span class="inline-marks">[2]</span></div></li></ol>
<p>The stall owner charges $\$10$ per game and rewards the player with a cash prize, in dollars, of $\frac{k}{10}$ times of the square of the player’s score, where&nbsp;$k$ is a positive integer.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1760836251407-obok3aau9" data-list-item-path="item-1760836251407-obok3aau9"><div class="list-item-paragraph">Determine the largest value of k for the stall to be profitable in the long run.<span class="inline-marks">[4]</span></div></li></ol>

At a funfair game stall, players are allowed to choose two cards at random from six cards, with each card labelled with one letter from A to F. The player’s score, denoted by $X$, is the Manhattan distance between the two squares corresponding to the player’s two chosen letters on the grid below,

where the Manhattan distance between two squares is the minimum total number of horizontal and vertical steps required to travel between them. For example, the Manhattan distance between B and F is $2$, while the Manhattan distance between D and C is $3$.

The stall owner charges $\$10$ per game and rewards the player with a cash prize, in dollars, of $\frac{k}{10}$ times of the square of the player’s score, where $k$ is a positive integer.

Expected Payoffs and Fair Games

2024 NJC P2 Q8

<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.

2024 TMJC P2 Q6

<p>In a certain airline route, there are a maximum of $250$ seats and each ticket is sold at $\$350$. If costs $\$1050$ for the airline for every overbooked passenger if they show up for the flight for compensations. Assume that a booked passenger will not show for the flight with probability $8\%$ and that the number of passengers who show up for each flight follows a binomial distribution. The airline has sold $254$ tickets for a particular flight.</p>
<ol class="abc-list"><li data-list-item-id="item-1760703015096-6nric3mxc" data-list-item-path="item-1760703015096-6nric3mxc"><div class="list-item-paragraph">What is the probability that there will not be enough seats for the flight?</div></li><li data-list-item-id="item-1760703016026-3n3fp80t6" data-list-item-path="item-1760703016026-3n3fp80t6"><div class="list-item-paragraph">Find the expected amount of revenue for selling $254$ tickets.</div></li></ol>
<p>Assume that the airline sells $n$ tickets with $n&gt;250$ as they decide to overbook.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-1760703030784-3c7yyg38d" data-list-item-path="item-1760703030784-3c7yyg38d"><div class="list-item-paragraph"><p>Find the expected amount of revenue for selling $n$ tickets in the form of $\mathrm{f}\left( n \right)\mathrm{P}\left( X\le 250 \right)+\sum\limits_{r=251}^{n}{\left( \left[ \mathrm{f}\left( n \right)-\mathrm{g}\left( r \right) \right]P\left( X=r \right) \right)}$, where&nbsp;$\mathrm{f}\left( n \right)$ and $\mathrm{g}\left( r \right)$ are functions in terms of $n$ and $r$ respectively to be determine.</p></div></li><li data-list-item-id="item-1760703031476-r55hw6fh6" data-list-item-path="item-1760703031476-r55hw6fh6"><div class="list-item-paragraph">Hence, find the number of tickets to be overbooked in order to maximise the revenue.</div></li></ol>

In a certain airline route, there are a maximum of $250$ seats and each ticket is sold at $\$350$. If costs $\$1050$ for the airline for every overbooked passenger if they show up for the flight for compensations. Assume that a booked passenger will not show for the flight with probability $8\%$ and that the number of passengers who show up for each flight follows a binomial distribution. The airline has sold $254$ tickets for a particular flight.

Assume that the airline sells $n$ tickets with $n>250$ as they decide to overbook.

Binomial Probabilities Using Technology

Prelims Revision Set: Problems involving Overbooking Q2

<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

J2 Prelims Revision Set Problems Involving Lengths of Objects Q1

<p>A discrete random variable $X$ can only take two possible values, $0$ and $1$. Find its maximum possible variance.</p>

A discrete random variable $X$ can only take two possible values, $0$ and $1$. Find its maximum possible variance.

J2 Prelims Revision Set Discrete Random Variable Q1

<p>Two fair dice are thrown and the product of the two numbers on the top faces is computed. Let&nbsp;${{R}_{n}}$ be the remainder when the product is divided by $n$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the probability distribution of ${{R}_{2}}$ and hence find $\mathrm{E}\left( {{R}_{2}} \right)$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Show that&nbsp;$\mathrm{P}\left( {{R}_{6}}=0 \right)=\frac{5}{12}$.<span class="inline-marks">[1]</span></div></li></ol>
<p>The two dice are thrown m times and the product of the two numbers on the top faces are computed each time. $X$ is the number of products that are divisible by $6$ after $m$ throws.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Show that&nbsp;$\mathrm{P}\left( X=2 \right)=Am\left( m-1 \right){{\left( \frac{7}{12} \right)}^{m}}$, where&nbsp;$A$ is a constant to be determined. Given that x$\mathrm{P}\left( X=2 \right)=\mathrm{0}\mathrm{.03604}$, correct to $5$ decimal places, deduce the value of $m$, justifying your answer.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Class $C$ has&nbsp;$35$ students and class $D$ has&nbsp;$30$ students. Each student in the two classes throws the two dice $10$ times. Estimate the probability that the average number of products that are divisible by $6$ obtained by a student in class $C$ exceeds the average number of products that are divisible by $6$ obtained by a student in class $D$ by more than $0.2$.<span class="inline-marks">[5]</span></div></li></ol>

Two fair dice are thrown and the product of the two numbers on the top faces is computed. Let ${{R}_{n}}$ be the remainder when the product is divided by $n$.

The two dice are thrown m times and the product of the two numbers on the top faces are computed each time. $X$ is the number of products that are divisible by $6$ after $m$ throws.

Binomial Probabilities Using the Probability Mass Function

Comparing Independent Sample Means

2023 RI P2 Q10

<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 $2.56$.

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