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>A letter is selected at random from each of the two words ‘SUN’ and ‘MOON’.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Construct a possibility diagram showing all the possible outcomes.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that the two letters are</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">not the same,</span></div></li><li><div class="list-item-content"><span class="list-item-text">vowels.</span></div></li></ol></li></ol>

A letter is selected at random from each of the two words ‘SUN’ and ‘MOON’.

Construct a possibility diagram showing all the possible outcomes.

Find the probability that the two letters are

Probability Using Possibility Diagrams

Probability from Letter Possibility Diagrams

Probability

<p>A fair die has the numbers $1$ to $6$ and a second fair die has the numbers $7$ to $12$. If the two dice are thrown, the score is the last digit in the product of the numbers shown on top of the dice.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Complete the possibility diagram below.</span></div></li></ol>
<div class="content-block content-block--table"><div class="table-container"><table><colgroup><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"></colgroup><thead><tr style="height: 43px;"><th style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$1$</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$2$</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$3$</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$4$</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$5$</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$6$</th></tr></thead><tbody><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$7$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$7$</td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$8$</td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$6$</td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$9$</td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$7$</td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$10$</td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$0$</td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$11$</td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$5$</td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$12$</td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"></td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$2$</td></tr></tbody></table></div></div>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the probability of getting a score that is</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">a perfect square,</span></div></li><li><div class="list-item-content"><span class="list-item-text">a prime number or a multiple of $3$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">a common factor of $30$ and $42$.</span></div></li></ol></li></ol>

A fair die has the numbers $1$ to $6$ and a second fair die has the numbers $7$ to $12$. If the two dice are thrown, the score is the last digit in the product of the numbers shown on top of the dice.

Find the probability of getting a score that is

Probability of Compound and Sequential Events

<p>A letter is selected at random from each of the two words 'EARTH' and 'MARS'.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Construct a possibility diagram showing all the possible outcomes.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the two letters are the same,</span></div></li><li><div class="list-item-content"><span class="list-item-text">at least one of the two letters is a vowel.</span></div></li></ol></li></ol>

A letter is selected at random from each of the two words 'EARTH' and 'MARS'.

Complementary Events

<p>Jerry has one $10$¢ coin, one $20$¢ coin, one $50$¢ coin and one $\$1$ coin in his right pocket. He has one $20$¢ coin, one $50$¢ coin and one $\$1$ coin in his left pocket. A coin is taken out at random from each pocket. Using a possibility diagram, find the probability that</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the total value of the two coins is $\$1.20$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the total value of the two coins is more than $\$1.20$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the two coins do not have the same value.</span></div></li></ol>

Jerry has one $10$¢ coin, one $20$¢ coin, one $50$¢ coin and one $\$1$ coin in his right pocket. He has one $20$¢ coin, one $50$¢ coin and one $\$1$ coin in his left pocket. A coin is taken out at random from each pocket. Using a possibility diagram, find the probability that

the total value of the two coins is $\$1.20$,

the total value of the two coins is more than $\$1.20$,

the two coins do not have the same value.

<p>A $6$-sided fair die is tossed first and a ball is drawn at random from a bag containing $2$ blue balls, $1$ yellow ball and $1$ green ball. The results are recorded in a possibility diagram.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Complete the possibility diagram shown below.</span></div></li></ol>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 560px;"><colgroup><col style="width: 40px;"><col style="width: 130px;"><col style="width: 130px;"><col style="width: 130px;"><col style="width: 130px;"></colgroup><tbody><tr style="height: 23px;"><td rowspan="2" style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Die</td><td colspan="4" style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Ball</p></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Blue</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Blue</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Yellow</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Green</p></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$1$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$1, B$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$1, B$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$1, Y$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$1, G$</p></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$2$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$2, B$</p></td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$3$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$3, B$</p></td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$4$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$4, B$</p></td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$5$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$5, B$</p></td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$6$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$6, B$</p></td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr></tbody></table></div></div>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the probability that the die shows</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">an even number and the ball drawn is yellow.</span></div></li><li><div class="list-item-content"><span class="list-item-text">a number less than $4$ and the ball drawn is not blue.</span></div></li></ol></li></ol>

A $6$-sided fair die is tossed first and a ball is drawn at random from a bag containing $2$ blue balls, $1$ yellow ball and $1$ green ball. The results are recorded in a possibility diagram.

Complete the possibility diagram shown below.

Constructing and Interpreting Two-Dimensional Possibility Diagrams

<p>A two-digit number is formed at random in a single event using the digits ${3, 2}$ and $4$ with repetition of digits allowed.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Write down the sample space.</div></li><li><div class="list-item-paragraph">Find the probability that the two-digit number</div><ol class="i-list"><li><div class="list-item-paragraph">contains the digit $3$,</div></li><li><div class="list-item-paragraph">does not contain the digit $3$,</div></li><li><div class="list-item-paragraph">is a prime.</div></li></ol></li></ol>

A two-digit number is formed at random in a single event using the digits ${3, 2}$ and $4$ with repetition of digits allowed.

Probability with Digit Formation

<p>Each of the numbers ${1, 2, 3}$ and $5$ is written on a card. Two cards are drawn uniformly at random in succession without replacement; the first gives the tens digit and the second the units digit. Find the probability that the two-digit number is </p>
<ol class="i-list"><li data-list-item-id="item-1765268759979-jvv8azaku" data-list-item-path="item-1765268759979-jvv8azaku"><div class="list-item-paragraph">divisible by $4$,</div></li><li data-list-item-id="item-1765268761441-nbexc4914" data-list-item-path="item-1765268761441-nbexc4914"><div class="list-item-paragraph">a prime number,</div></li><li data-list-item-id="item-1765268762049-d7mxxzqac" data-list-item-path="item-1765268762049-d7mxxzqac"><div class="list-item-paragraph">not divisible by $5$. </div></li></ol>

Each of the numbers ${1, 2, 3}$ and $5$ is written on a card. Two cards are drawn uniformly at random in succession without replacement; the first gives the tens digit and the second the units digit. Find the probability that the two-digit number is 

Probability Forming a Two-Digit Number

<p>In a game, Alex and Ben take turns to throw a pair of dice. Alex wins the game if the sum of two dice is $9$; Ben wins if the sum is $7$. The game stops when one of them wins the game. If Alex starts the game, find the probability of winning the game by Ben.</p>

In a game, Alex and Ben take turns to throw a pair of dice. Alex wins the game if the sum of two dice is $9$; Ben wins if the sum is $7$. The game stops when one of them wins the game. If Alex starts the game, find the probability of winning the game by Ben.

Repeated Trials with Infinitely Many Possible Outcomes

Probabilities of Successive Events

Waiting-Time and Stopping-Rule Probabilities

Infinite Outcomes Q1

<p>In a game played by two people $A$ and $B$, each player is given three cards with three different animal pictures printed on them. The players flash their cards simultaneously to indicate one of the three animals, ‘elephant’, ‘cat’ and ‘mouse’. ‘Elephant’ defeats ‘cat’, ‘cat’ defeats ‘mouse’ and ‘mouse’ defeats ‘elephant’. If the animals are the same, the game is a draw. At each game, $A$ indicates ‘elephant’, ‘cat’ and ‘mouse’ with probabilities of $0.3$, $0.3$ and $0.4$ respectively, while the corresponding probabilities for $B$ are $0.2$, $0.5$ and $0.3$. The game continues until a winner is found.</p>
<p>Find the probability that</p>
<ol class="i-list"><li><div class="list-item-paragraph">a game ends in a draw.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">$A$ is the winner in the first game;</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">$B$ is the winner in the first game given that the first game does not end in a draw;</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">$B$ will be the winner in the contest.</div><p style="text-align: right;">[2]</p></li></ol>

In a game played by two people $A$ and $B$, each player is given three cards with three different animal pictures printed on them. The players flash their cards simultaneously to indicate one of the three animals, ‘elephant’, ‘cat’ and ‘mouse’. ‘Elephant’ defeats ‘cat’, ‘cat’ defeats ‘mouse’ and ‘mouse’ defeats ‘elephant’. If the animals are the same, the game is a draw. At each game, $A$ indicates ‘elephant’, ‘cat’ and ‘mouse’ with probabilities of $0.3$, $0.3$ and $0.4$ respectively, while the corresponding probabilities for $B$ are $0.2$, $0.5$ and $0.3$. The game continues until a winner is found.

Conditional Probability

2008 JJC P2 Q9

<p>In a game played by two people A and B, each player is given three cards with three different animal pictures printed on them. The players flash their cards simultaneously to indicate one of the three animals, ‘elephant’, ‘cat’, and ‘mouse’. ‘Elephant’ defeats ‘cat’, ‘cat’ defeats ‘mouse’ and ‘mouse’ defeats ‘elephant’. If the animals are the same, the round is a draw. At each round, A indicates ‘elephant’, ‘cat’, and ‘mouse’ with probabilities $0.3$, $0.3$ and $0.4$ respectively, while the corresponding probabilities for B are $0.2$, $0.5$ and $0.3$. The game continues until a winner is found. Find the probability that</p>
<ol class="i-list"><li><div class="list-item-paragraph">a round ends in a draw,</div></li><li><div class="list-item-paragraph">A is the winner in the first round,</div></li><li><div class="list-item-paragraph">B is the winner in the first round given that the first round does not end in a draw,</div></li><li><div class="list-item-paragraph">B will be the winner in the contest.</div></li></ol>

In a game played by two people A and B, each player is given three cards with three different animal pictures printed on them. The players flash their cards simultaneously to indicate one of the three animals, ‘elephant’, ‘cat’, and ‘mouse’. ‘Elephant’ defeats ‘cat’, ‘cat’ defeats ‘mouse’ and ‘mouse’ defeats ‘elephant’. If the animals are the same, the round is a draw. At each round, A indicates ‘elephant’, ‘cat’, and ‘mouse’ with probabilities $0.3$, $0.3$ and $0.4$ respectively, while the corresponding probabilities for B are $0.2$, $0.5$ and $0.3$. The game continues until a winner is found. Find the probability that

2008 JJC P2 Q7

<p>Two fair dice, one red and the other green, are thrown.</p>
<p style="text-align: center">$A$ is the event: The score on the red die is divisible by $3$.</p>
<p style="text-align: center">$B$ is the event: The sum of the two scores is $9$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Justifying your conclusion, determine whether $A$ and $B$ are independent.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">The experiment is repeated multiple times. How many times must the experiment be conducted, so that the probability that&nbsp;$B$ occurs at least once exceeds $0.98$?<span class="inline-marks">[3]</span></div></li></ol>

Two fair dice, one red and the other green, are thrown.

$A$ is the event: The score on the red die is divisible by $3$.

$B$ is the event: The sum of the two scores is $9$.

Justifying your conclusion, determine whether $A$ and $B$ are independent.

The experiment is repeated multiple times. How many times must the experiment be conducted, so that the probability that $B$ occurs at least once exceeds $0.98$?

Independent Events

Minimum Number of Binomial Trials

N2006 P2 Q23 [Modified]

Discrete Random Variables (DRV)

<p>A fair die and a biased die are thrown. The events $A$, $B$ and $C$ are defined as follows:</p>
<p style="text-align: center">$A$: the score on the fair die is odd</p>
<p style="text-align: center">$B$: the score on the biased die is even</p>
<p style="text-align: center">$C$: the sum of the two scores is odd.</p>
<p>The biased die has probabilities given by $\mathrm{P}$ (score is $r$) $=\frac{r}{21}$ where $r=1,2,3,4,5,6$. Identify two events that are independent. Support your answers with a reason.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find $\mathrm{P}\left( A \right)$, $\mathrm{P}\left( B \right)$ and $\mathrm{P}\left( C \right)$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find $\mathrm{P}\left( A\cap C \right)$ and $\mathrm{P}\left( A|C \right)$.</span></div></li></ol>

A fair die and a biased die are thrown. The events $A$, $B$ and $C$ are defined as follows:

The biased die has probabilities given by $\mathrm{P}$ (score is $r$) $=\frac{r}{21}$ where $r=1,2,3,4,5,6$. Identify two events that are independent. Support your answers with a reason.

Find $\mathrm{P}\left( A \right)$, $\mathrm{P}\left( B \right)$ and $\mathrm{P}\left( C \right)$.

Find $\mathrm{P}\left( A\cap C \right)$ and $\mathrm{P}\left( A|C \right)$.

Probability Q1: A Fair and a Biased Die are Thrown

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Suppose a fair die is tossed twice. Calculate the probabilities that</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the sum of the scores of the two tosses is at least $8$,</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">the absolute difference between the scores of the two tosses is at least $4$.</span></div><p style="text-align: right;">[1]</p></li></ol></li></ol>
<p>In one round of a game, a player is to draw a ball, without replacement, from a box that contains $3$ red balls and $4$ white balls. If a red ball is drawn, the player will add the scores obtained from tossing a fair die twice. If a white ball is drawn, the player will take the absolute difference of the scores obtained from tossing a die twice.</p>
<p>The game ends if the sum of the scores is at least 8 or the absolute difference of the scores is at least $4$. Else, the player will proceed to the second round of the game where the process of picking a ball from the box and tossing the die twice repeats.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the probability that the game ends at the first round.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Suppose the game ends at the first round. Find the probability that a red ball is drawn.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that there are a total of $3$ rounds of game played and exactly $2$ white balls are selected.</span></div><p style="text-align: right;">[3]</p></li></ol>

Suppose a fair die is tossed twice. Calculate the probabilities that

In one round of a game, a player is to draw a ball, without replacement, from a box that contains $3$ red balls and $4$ white balls. If a red ball is drawn, the player will add the scores obtained from tossing a fair die twice. If a white ball is drawn, the player will take the absolute difference of the scores obtained from tossing a die twice.

The game ends if the sum of the scores is at least 8 or the absolute difference of the scores is at least $4$. Else, the player will proceed to the second round of the game where the process of picking a ball from the box and tossing the die twice repeats.

Find the probability that the game ends at the first round.

Suppose the game ends at the first round. Find the probability that a red ball is drawn.

Find the probability that there are a total of $3$ rounds of game played and exactly $2$ white balls are selected.

Bayes' Theorem

Law of Total Probability

Conditional Probability from Tree Diagrams

2016 NJC P2 Q10

<p>An experiment is carried out with three coins. Two of the coins are fair. The third coin is biased such that the probability of obtaining a head is $\frac{1}{5}$. The three coins are tossed.</p>
<p style="margin-left: 11rem">$A$ is the event: All the three coins show the same result.<br>$B$ is the event: The biased coin shows a tail.</p>
<p>Find</p>
<ol class="i-list"><li><div class="list-item-paragraph">${P(A \cup B)}$,<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">${P(B'|A')}$.<span class="inline-marks">[3]</span></div></li></ol>

An experiment is carried out with three coins. Two of the coins are fair. The third coin is biased such that the probability of obtaining a head is $\frac{1}{5}$. The three coins are tossed.

$A$ is the event: All the three coins show the same result.
$B$ is the event: The biased coin shows a tail.

Addition Rule for Overlapping Events

2016 AJC P2 Q6

<p>Two fair dice are thrown simultaneously. Events $A$ and $B$ are defined as follows.</p>
<p style="margin-left: 3rem">$A$: $\quad$ at least one six appears,</p>
<p style="margin-left: 3rem">$B$: $\quad$ the sum of two scores is ten.</p>
<p>Find</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\mathrm{P}\left( A|B \right)$,<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">$\mathrm{P}\left( A|B' \right)$.<span class="inline-marks">[2]</span></div></li></ol>
<p>For a third event $C$, it is given that&nbsp;$\mathrm{P}\left( B'\cup C \right)=\frac{35}{36}$ and that&nbsp;$B$ and $C$ are independent.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the exact value of $\mathrm{P}\left( C \right)$.<span class="inline-marks">[3]</span></div></li></ol>

Two fair dice are thrown simultaneously. Events $A$ and $B$ are defined as follows.

$B$: $\quad$ the sum of two scores is ten.

For a third event $C$, it is given that $\mathrm{P}\left( B'\cup C \right)=\frac{35}{36}$ and that $B$ and $C$ are independent.

Find the exact value of $\mathrm{P}\left( C \right)$.

Math Question Bank