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<p>Players $A$ and $B$ compete in a racquet match consisting of at most $3$ sets. Each set is won by either Player $A$ or $B$, and the match is won by the first person to win two sets. Player $A$ has a probability of $\frac{2}{3}$ of winning each of the first two sets. If the match goes into the third set, Player $A$ has a probability of $p$ of winning this set.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">With the aid of a tree diagram, find in terms of $p$, the probability that player $A$ will win the match.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Deduce the range of values of the probability that player $A$ wins the match.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that the value of $p$ in order for the match to be fair is $\frac{1}{8}$.</span></div></li></ol>
<p>Given that Player $A$ wins the match, find, in terms of $p$, the probability that he won the $2^{\text{nd}}$ set.</p>

Players $A$ and $B$ compete in a racquet match consisting of at most $3$ sets. Each set is won by either Player $A$ or $B$, and the match is won by the first person to win two sets. Player $A$ has a probability of $\frac{2}{3}$ of winning each of the first two sets. If the match goes into the third set, Player $A$ has a probability of $p$ of winning this set.

With the aid of a tree diagram, find in terms of $p$, the probability that player $A$ will win the match.

Deduce the range of values of the probability that player $A$ wins the match.

Show that the value of $p$ in order for the match to be fair is $\frac{1}{8}$.

Given that Player $A$ wins the match, find, in terms of $p$, the probability that he won the $2^{\text{nd}}$ set.

Finding Unknowns in Probability Models

Waiting-Time and Stopping-Rule Probabilities

2011 NYJC Prelim

Probability

<p>Two events $A$ and $B$ are such that&nbsp;$\mathrm{P}(A) = p$ and $\mathrm{P}(B) = \frac{5}{4}p$. It is given that&nbsp;$\mathrm{P}(A \cup B) = \mathrm{P}(A|B) = 0.6$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the value of $p$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain whether the events $A'$ and $B$ are independent.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>The events $C$ and $D$ are such that&nbsp;$\mathrm{P}(C) = 0.55$, $\mathrm{P}(D) = 0.6$ and $\mathrm{P}(A \cap C) = \mathrm{P}(A \cap D) = \mathrm{P}(C \cap D) = q$.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the value of $q$ that gives the minimum value of $\mathrm{P}(A \cup C \cup D)$ and state the minimum value of $\mathrm{P}(A \cup C \cup D)$.</span><span class="inline-marks">[2]</span></div></li></ol>

Two events $A$ and $B$ are such that $\mathrm{P}(A) = p$ and $\mathrm{P}(B) = \frac{5}{4}p$. It is given that $\mathrm{P}(A \cup B) = \mathrm{P}(A|B) = 0.6$.

Explain whether the events $A'$ and $B$ are independent.

The events $C$ and $D$ are such that $\mathrm{P}(C) = 0.55$, $\mathrm{P}(D) = 0.6$ and $\mathrm{P}(A \cap C) = \mathrm{P}(A \cap D) = \mathrm{P}(C \cap D) = q$.

Find the value of $q$ that gives the minimum value of $\mathrm{P}(A \cup C \cup D)$ and state the minimum value of $\mathrm{P}(A \cup C \cup D)$.

Addition Rule for Overlapping Events

2025 VJC P2 Q6

<p>A bag contains $15$ tokens that are indistinguishable apart from their colours. $2$ of the tokens are blue and the rest are either red or green. Participants are required to draw the tokens randomly, one at a time, from the bag without replacement.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that the probability that a participant draws $2$ red tokens on the first $2$ draws is $\frac{1}{35}$, show that there are $3$ red tokens in the bag.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the probability that a participant draws a red or green token on the second draw.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Events $A$ and $B$ are defined as follows:</div><div class="list-item-paragraph">$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ $A$: A participant draws his/her second red token on the third draw.<br>$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ $B$: A participant draws a blue token on the second draw.</div><div class="list-item-paragraph">Find $\mathrm{P}(A \cup B)$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Determine if&nbsp;$A$ and $B$ are independent events.<span class="inline-marks">[1]</span></div></li></ol>

A bag contains $15$ tokens that are indistinguishable apart from their colours. $2$ of the tokens are blue and the rest are either red or green. Participants are required to draw the tokens randomly, one at a time, from the bag without replacement.

Given that the probability that a participant draws $2$ red tokens on the first $2$ draws is $\frac{1}{35}$, show that there are $3$ red tokens in the bag.

Find the probability that a participant draws a red or green token on the second draw.

Independent Events

Probabilities of Successive Events

Probability Using Permutations and Combinations

2013 HCI P2 Q10

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">If&nbsp;$\mathrm{P}(B|A) = 0.2$, $\mathrm{P}(A|B) = 0.6$ and $\mathrm{P}(A' \cap B') = 0.3$, find $\mathrm{P}(A \cap B)$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">The events $X$, $Y$ and $Z$ are such that the events $X$ and $Z$ are mutually exclusive. Given further that&nbsp;$\mathrm{P}(X \cap Y) = 0.1$, $\mathrm{P}(X' \cap Y) = 0.35$, $\mathrm{P}(Y \cap Z) = 0.2$, and $\mathrm{P}(X \cup Y \cup Z) = 0.95$, find the minimum and maximum values of $\mathrm{P}(X)$.</span><span class="inline-marks">[3]</span></div></li></ol>

If $\mathrm{P}(B|A) = 0.2$, $\mathrm{P}(A|B) = 0.6$ and $\mathrm{P}(A' \cap B') = 0.3$, find $\mathrm{P}(A \cap B)$.

The events $X$, $Y$ and $Z$ are such that the events $X$ and $Z$ are mutually exclusive. Given further that $\mathrm{P}(X \cap Y) = 0.1$, $\mathrm{P}(X' \cap Y) = 0.35$, $\mathrm{P}(Y \cap Z) = 0.2$, and $\mathrm{P}(X \cup Y \cup Z) = 0.95$, find the minimum and maximum values of $\mathrm{P}(X)$.

2025 ACJC P2 Q6

<p>A bag has marbles of three different colours: red, blue and white. Two marbles are to be drawn from the bag, one after another without replacement. A partially completed tree diagram is shown below.<br>Given that the probability of drawing two marbles of <strong>different</strong> colours is $\frac{31}{45}$, find</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="list-item-paragraph">the probability of drawing two marbles of the <strong>same</strong> colour,</div></li><li><div class="list-item-paragraph">the value of $x$,</div></li><li><div class="list-item-paragraph">the value of $y$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767698701664-1tvbqa1h45w.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTg3MDE2NjQtMXR2YnFhMWg0NXcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTc4MSwiZXhwIjoxNzkyNzUwMDgxfQ.MR6HU4DZzynDkTCaL5s0JMDQDRg80jB78EGovIMTtT8" style="max-width: min(290px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

A bag has marbles of three different colours: red, blue and white. Two marbles are to be drawn from the bag, one after another without replacement. A partially completed tree diagram is shown below.
Given that the probability of drawing two marbles of **different** colours is $\frac{31}{45}$, find

Complementary Events

Dependent Events

Constructing and Using Probability Trees

Theoretical Probability in Successive Experiments

<p>There are $33$ red balls, $x$ white balls and $y$ black balls in a box. When one ball is drawn at random from the box, the probability that the ball is white is $0.2$ and the probability that the ball is black is $0.25$. Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the probability that the ball is red,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the values of $x$ and $y$.</span></div></li></ol>

There are $33$ red balls, $x$ white balls and $y$ black balls in a box. When one ball is drawn at random from the box, the probability that the ball is white is $0.2$ and the probability that the ball is black is $0.25$. Find

Probability Analysis: Determining Unknown Frequencies via Algebraic Equations

<p>There are $12$ girls and $x$ boys in a group. The probability of selecting a boy randomly from the group is $\frac{3}{7}$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the value of $x$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">How many more boys are needed to join the group so that the probability of selecting a boy randomly from the group will be $\frac{5}{6}$?</span></div></li></ol>

There are $12$ girls and $x$ boys in a group. The probability of selecting a boy randomly from the group is $\frac{3}{7}$.

How many more boys are needed to join the group so that the probability of selecting a boy randomly from the group will be $\frac{5}{6}$?

Solving for Unknown Frequencies in Probability Sets

<p>A standard pack of $52$ playing cards is randomly divided into two unequal piles. Given that the probability of drawing a picture card from the smaller pile is $\frac{4}{11}$ while the probability of drawing a non-picture card from the bigger pile is $\frac{13}{15}$, find the number of cards in each pile.</p>

A standard pack of $52$ playing cards is randomly divided into two unequal piles. Given that the probability of drawing a picture card from the smaller pile is $\frac{4}{11}$ while the probability of drawing a non-picture card from the bigger pile is $\frac{13}{15}$, find the number of cards in each pile.

Probability: Selecting Cards from Two Piles

<p>There are $50$ students in an auditorium, of which $2x$ are boys and $y$ are girls. After $\left( y-6 \right)$ boys leave the auditorium and $\left( 2x-5 \right)$ girls enter the auditorium, the probability of selecting a girl at random becomes&nbsp;$\frac{9}{13}$. Find the value of $x$ and of $y$.</p>

There are $50$ students in an auditorium, of which $2x$ are boys and $y$ are girls. After $\left( y-6 \right)$ boys leave the auditorium and $\left( 2x-5 \right)$ girls enter the auditorium, the probability of selecting a girl at random becomes $\frac{9}{13}$. Find the value of $x$ and of $y$.

Probability: Selecting a Student from a Group (Algebraic Form)

<p>A bag contains $40$ balls, some of which are blue, some of which are green, and the rest are white. The probabilities of drawing a blue ball and a green ball at random from the bag are $\frac{1}{4}$ and $\frac{2}{5}$ respectively.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the probability of drawing a white ball at random from the bag.<br>$\left( 2x+1 \right)$ blue balls and $\left( x+2 \right)$ green balls are added to the bag while $\left( x-3 \right)$ white balls are removed from the bag. The probability of drawing a green ball at random from the bag is now $\frac{3}{7}$. Find<br></div></li><li><div class="list-item-paragraph">an expression, in terms of $x$, for the total number of balls in the bag now,</div></li><li><div class="list-item-paragraph">the number of green balls in the bag now.</div></li></ol>

A bag contains $40$ balls, some of which are blue, some of which are green, and the rest are white. The probabilities of drawing a blue ball and a green ball at random from the bag are $\frac{1}{4}$ and $\frac{2}{5}$ respectively.

Probability: Drawing Balls from a Bag (Algebraic Form)

<p>There were initially $48$ boys and $2$ girls in a basketball team. After a sports fair, $x$ boys and $2x$ girls joined the team. Given that the probability of selecting a boy at random from the team is now $\frac{2}{5}$, find the number of girls that joined the team.</p>

There were initially $48$ boys and $2$ girls in a basketball team. After a sports fair, $x$ boys and $2x$ girls joined the team. Given that the probability of selecting a boy at random from the team is now $\frac{2}{5}$, find the number of girls that joined the team.

Probability: Selecting a Student from a Team

<p>A tank contains $50$ fishes of equal size of which $18$ are green, $x$ are yellow and the rest are red.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">If a fish is selected randomly from the tank, find the probability that the fish selected is</div><ol class="i-list"><li><div class="list-item-paragraph">green,</div></li><li><div class="list-item-paragraph">not green.</div></li></ol></li><li><div class="list-item-paragraph">If the probability of selecting a yellow fish is $\frac{3}{10}$, find the value of $x$.</div></li></ol>

A tank contains $50$ fishes of equal size of which $18$ are green, $x$ are yellow and the rest are red.

Probability of a Single Event

Probability: Choosing a Fish from a Tank

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765333907233-pmxyzzhpzai.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUzMzM5MDcyMzMtcG14eXp6aHB6YWkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTc4MSwiZXhwIjoxNzkyNzUwMDgxfQ.iBGmGm9RrAqBfxULeDEHpSNYsyrDaFxceqrgtDcnnCo" style="max-width: min(200px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The pie chart above shows a circle divided into sectors of different colours.  A point is selected randomly in the circle.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Calculate the probability that the point lies in the Red sector.</div></li><li><div class="list-item-paragraph">Calculate the probability that the point does not lie in the Blue sector.</div></li><li><div class="list-item-paragraph">Given the probability that the point lies in the Yellow sector is $\frac{3}{10}$, find the value of $x$.</div></li><li><div class="list-item-paragraph">Hence, calculate the probability that the point lies in the Black sector.</div></li></ol>

The pie chart above shows a circle divided into sectors of different colours.  A point is selected randomly in the circle.

Probability: Reading a Sector Diagram

<p>In a certain school, students are grouped into Red house, Blue house and Green house. There is a group of students from different houses gathered in the school hall.  A student is selected at random from the group.  The probability of selecting a student from the Red house is $\frac{1}{3}$.  The probability of selecting a student from the Blue house is $\frac{3}{8}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the probability of selecting a student from Green house.</div></li><li><div class="list-item-paragraph">If there are 24 students from Blue house, how many students are there in the group?<br><br></div></li></ol>

In a certain school, students are grouped into Red house, Blue house and Green house. There is a group of students from different houses gathered in the school hall.  A student is selected at random from the group.  The probability of selecting a student from the Red house is $\frac{1}{3}$.  The probability of selecting a student from the Blue house is $\frac{3}{8}$.

Probability: Choosing a Student from a Group

<p>A ball is selected at random from a bag. The bag contains $30$ balls of which $8$ balls are blue, $x$ balls are red and the rest are purple. </p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the probability of selecting </div><ol class="i-list"><li><div class="list-item-paragraph">a blue ball,</div></li><li><div class="list-item-paragraph">a black ball.</div></li></ol></li><li><div class="list-item-paragraph">If the probability of picking a purple ball is $\frac{1}{5}$, find the value of $x$.</div></li></ol>

A ball is selected at random from a bag. The bag contains $30$ balls of which $8$ balls are blue, $x$ balls are red and the rest are purple. 

Probability: Picking a Ball from a Bag

<p>A deck of cards consists of 3 suits. The deck contains $\left( 24-x \right)$ cards, $\left( 5x-8 \right)$ cards and $\left( 2x-7 \right)$ cards with the suits of clubs, hearts and spades respectively. </p>
<ol class="i-list"><li><div class="list-item-paragraph">Find an expression, in terms of $x$, for the total number of cards in the deck. </div></li></ol>
<p>A card is drawn at random from the box. </p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Write down an expression, in terms of $x$, for the probability that the card has the suit of clubs.</div></li><li><div class="list-item-paragraph">It is known that the probability when a card with the suit of clubs is not drawn is $\frac{2}{3}$, find the value of $x$.</div></li></ol>

A deck of cards consists of 3 suits. The deck contains $\left( 24-x \right)$ cards, $\left( 5x-8 \right)$ cards and $\left( 2x-7 \right)$ cards with the suits of clubs, hearts and spades respectively. 

Probability: Selecting a Card from a Variable Deck

<p>A deck of cards consists of $3$ suits. The deck contains $\left( 6x-8 \right)$ cards, $\left( 25-2x \right)$ cards and $\left( 4x+3 \right)$ cards with the suits of clubs, hearts, and spades respectively. </p>
<ol class="i-list"><li><div class="list-item-paragraph">Find an expression, in terms of $x$, for the total number of cards in the deck.</div></li></ol>
<p>A card is drawn at random from the box. </p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Write down an expression, in terms of $x$, for the probability that the card has the suit of spades.</div></li><li><div class="list-item-paragraph">It is known that the probability when a card with the suit of spades is <strong>not</strong> drawn is $\frac{7}{12}$,	find the value of $x$.</div></li></ol>

A deck of cards consists of $3$ suits. The deck contains $\left( 6x-8 \right)$ cards, $\left( 25-2x \right)$ cards and $\left( 4x+3 \right)$ cards with the suits of clubs, hearts, and spades respectively. 

Probability in a Three-Suit Deck

<p>A box contains $\left( 6y+2 \right)$ red balls, $\left( 50-2y \right)$ blue balls and $3y$ yellow balls. </p>
<ol class="i-list"><li><div class="list-item-paragraph">Find an expression, in terms of $y$, for the total number of balls in the box. </div></li></ol>
<p>A ball is drawn at random from the box. </p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Write down an expression, in terms of $y$, for the probability that the ball is red. </div></li><li><div class="list-item-paragraph">It is known that the probability when a red ball is drawn is $\frac{13}{40}$, find the value of $y$.</div></li></ol>

A box contains $\left( 6y+2 \right)$ red balls, $\left( 50-2y \right)$ blue balls and $3y$ yellow balls. 

Balls in a Box: Variable-Based Probability

<p>A box contains $x$ red balls, $\left( 2x-3 \right)$ green balls and $\left( 22-4x \right)$ blue balls. </p>
<ol class="i-list"><li><div class="list-item-paragraph">Find an expression, in terms of $x$, for the total number of balls in the box. </div></li></ol>
<p>A ball is drawn at random from the box. </p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Write down an expression, in terms of $x$, for the probability that the ball is blue. </div></li><li><div class="list-item-paragraph">It is known that the probability when a blue ball is drawn is $\frac{5}{8}$, find the value of $x$. </div></li></ol>

A box contains $x$ red balls, $\left( 2x-3 \right)$ green balls and $\left( 22-4x \right)$ blue balls. 

Probability: Drawing a Ball from a Box (Algebraic Form)

<p>A bag contains a number of black and white balls of same shape and size. A ball is drawn at random from the bag.  Let&nbsp;$x$ be the probability that the ball drawn is black and $y$ be the probability that the ball drawn is white.  If&nbsp;$x=4y$, find $x$.</p>

A bag contains a number of black and white balls of same shape and size. A ball is drawn at random from the bag.  Let $x$ be the probability that the ball drawn is black and $y$ be the probability that the ball drawn is white.  If $x=4y$, find $x$.

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