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>Two unbiased dice are in the form of a regular tetrahedron and each has its faces numbered $1$, $2$, $3$ and $4$. When each die is thrown on to a horizontal table, the number on the face in contact with the table is noted.</p>
<p>Suppose Tom and Harry play the following game: Tom throws the $2$ dice together and notes the total score, $X$. Harry tries to guess this total score and his guess is denoted by $Y$. If Harry guesses correctly, he wins $\$Y^{2}$, otherwise he loses $\$\frac{1}{2}Y$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Obtain the probability distribution of $X$ and find $\mathrm{E}\left(\frac{1}{X}\right)$, correct to $3$ decimal places.</div></li><li><div class="list-item-paragraph">Harry guesses the total score to be $6$ in a particular game. Find the expected gain in this game.</div></li></ol>

Two unbiased dice are in the form of a regular tetrahedron and each has its faces numbered $1$, $2$, $3$ and $4$. When each die is thrown on to a horizontal table, the number on the face in contact with the table is noted.

Suppose Tom and Harry play the following game: Tom throws the $2$ dice together and notes the total score, $X$. Harry tries to guess this total score and his guess is denoted by $Y$. If Harry guesses correctly, he wins $\$Y^{2}$, otherwise he loses $\$\frac{1}{2}Y$.

Expected Values of Functions of Discrete Random Variables

Expected Payoffs and Fair Games

<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 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 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.

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>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

2024 NYJC P2 Q7

<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>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 bag contains $5$ red balls and $3$ blue balls. In a game, Anne removes balls at random from the bag, one at a time, without replacement, until she has taken out $2$ red balls. The total number of balls Anne removes from the bag is denoted by $X$.</p>
<ol class="i-list"><li data-list-item-id="item-1760447515891-9o42id1y1" data-list-item-path="item-1760447515891-9o42id1y1"><div class="list-item-paragraph">Find $\mathrm{P}\left( X=x \right)$ for all possible values of $x$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760447516596-xp82o83m1" data-list-item-path="item-1760447516596-xp82o83m1"><div class="list-item-paragraph">Find $\mathrm{E}\left( X \right)$ and $\mathrm{Var}\left( X \right)$.<span class="inline-marks">[3]</span></div></li></ol>
<p>Anne pays $\$1$ every time she removes a ball and she will receive $\$y$ once she gets $2$ red balls.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760447535353-jh898eecr" data-list-item-path="item-1760447535353-jh898eecr"><div class="list-item-paragraph">Calculate the value of $y$ if the game is fair.<span class="inline-marks">[2]</span></div></li></ol>

A bag contains $5$ red balls and $3$ blue balls. In a game, Anne removes balls at random from the bag, one at a time, without replacement, until she has taken out $2$ red balls. The total number of balls Anne removes from the bag is denoted by $X$.

Anne pays $\$1$ every time she removes a ball and she will receive $\$y$ once she gets $2$ red balls.

Probabilities of Successive Events

Waiting-Time and Stopping-Rule Probabilities

2023 SAJC P2 Q6

<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>A bag contains $2$ fair tetrahedral dice. The first die has faces labelled $1$, $1$, $2$, and $3$ and the second die has faces labelled $1$, $2$, $3$ and $3$. A die is taken at random from the bag and thrown. The score, $W$ is defined as follows:</p>
<p>If the first die is picked and thrown, the score is defined as twice the number which appears on its base.</p>
<p>If the second die is picked and thrown, the score is the number which appears on its base.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $\mathrm{P}\left( W=2 \right)=\frac{3}{8}$ and find the probability distribution of $W$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find $\mathrm{E}\left( W \right)$ and $\mathrm{Var}\left( W \right)$.</div><p style="text-align: right;">[4]</p></li></ol>

A bag contains $2$ fair tetrahedral dice. The first die has faces labelled $1$, $1$, $2$, and $3$ and the second die has faces labelled $1$, $2$, $3$ and $3$. A die is taken at random from the bag and thrown. The score, $W$ is defined as follows:

If the first die is picked and thrown, the score is defined as twice the number which appears on its base.

If the second die is picked and thrown, the score is the number which appears on its base.

2022 JPJC P2 Q5

<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>Urn $1$ contains $5$ black balls and $3$ red balls. Urn $2$ contains $4$ black balls and $4$ red balls. An experiment is conducted in the following manner:</p>
<p>A fair die is tossed. If it shows $1$ or $2$, two balls are drawn from urn $1$ without replacement.</p>
<p>Otherwise, one ball is drawn from each urn. Let $X$ be the number of red balls drawn in one experiment.</p>
<p>Show that $\mathrm{P}\left( X=0 \right)=\frac{55}{168}$.</p><p style="text-align: right;">[1]</p>
<p>Find the probability distribution of $X$. Evaluate the expectation and variance of $X$.</p><p style="text-align: right;">[6]</p>

Urn $1$ contains $5$ black balls and $3$ red balls. Urn $2$ contains $4$ black balls and $4$ red balls. An experiment is conducted in the following manner:

A fair die is tossed. If it shows $1$ or $2$, two balls are drawn from urn $1$ without replacement.

Otherwise, one ball is drawn from each urn. Let $X$ be the number of red balls drawn in one experiment.

Show that $\mathrm{P}\left( X=0 \right)=\frac{55}{168}$.

Find the probability distribution of $X$. Evaluate the expectation and variance of $X$.

Math Question Bank