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PSLE

PSLE Foundation Mathematics

<p>In a group of $16$ students, $12$ take art and $8$ take music. One student takes neither art nor music. The Venn diagram below shows the events art and music. The values $p, q, r$ and $s$ represent numbers of students.</p>
<div class="graph-block" data-graph-config="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" data-width="520" data-height="260" style="width:520px;height:260px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="abc-list"><li data-list-item-id="ibsl-q002-part-1-014" data-list-item-path="ibsl-q002-part-1-014"><div class="list-item-paragraph"><span class="inline-marks">[5]</span></div><ol class="i-list"><li data-list-item-id="ibsl-q002-part-1-1-015" data-list-item-path="ibsl-q002-part-1-014/ibsl-q002-part-1-1-015"><div class="list-item-paragraph"><p>Write down the value of $s$.</p></div></li><li data-list-item-id="ibsl-q002-part-1-2-016" data-list-item-path="ibsl-q002-part-1-014/ibsl-q002-part-1-2-016"><div class="list-item-paragraph"><p>Find the value of $q$.</p></div></li><li data-list-item-id="ibsl-q002-part-1-3-017" data-list-item-path="ibsl-q002-part-1-014/ibsl-q002-part-1-3-017"><div class="list-item-paragraph"><p>Write down the value of $p$ and of $r$.</p></div></li></ol></li><li data-list-item-id="ibsl-q002-part-2-018" data-list-item-path="ibsl-q002-part-2-018"><div class="list-item-paragraph"><span class="inline-marks">[4]</span></div><ol class="i-list"><li data-list-item-id="ibsl-q002-part-2-1-019" data-list-item-path="ibsl-q002-part-2-018/ibsl-q002-part-2-1-019"><div class="list-item-paragraph">A student is selected at random. Given that the student takes music, write down the probability the student takes art.</div></li><li data-list-item-id="ibsl-q002-part-2-2-020" data-list-item-path="ibsl-q002-part-2-018/ibsl-q002-part-2-2-020"><div class="list-item-paragraph">Hence, show that taking music and taking art are not independent events.</div></li></ol></li><li data-list-item-id="ibsl-q002-part-3-021" data-list-item-path="ibsl-q002-part-3-021"><div class="list-item-paragraph">Two students are selected at random, one after the other. Find the probability that the first student takes only music and the second student takes only art.<span class="inline-marks">[4]</span></div></li></ol>

In a group of $16$ students, $12$ take art and $8$ take music. One student takes neither art nor music. The Venn diagram below shows the events art and music. The values $p, q, r$ and $s$ represent numbers of students.

Two students are selected at random, one after the other. Find the probability that the first student takes only music and the second student takes only art.

Set Theory and Probability — Combined Problems

Independent Events

Conditional Probability

Dependent Events

Complete a Venn diagram, calculate conditional probability, test independence and sample without replacement.

Probability

<p>Let&nbsp;$A$ and $B$ be independent events, where&nbsp;$P(A) = 0.6$ and $P(B) = x$.</p>
<ol class="abc-list"><li data-list-item-id="ibsl-q033-part-1-003" data-list-item-path="ibsl-q033-part-1-003"><div class="list-item-paragraph">Write down an expression for&nbsp;$P(A ∩ B)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="ibsl-q033-part-2-004" data-list-item-path="ibsl-q033-part-2-004"><div class="list-item-paragraph"><p>Given that&nbsp;$P(A ∪ B) = 0.8$,</p><span class="inline-marks">[4]</span></div><ol class="i-list"><li data-list-item-id="ibsl-q033-part-2-1-005" data-list-item-path="ibsl-q033-part-2-004/ibsl-q033-part-2-1-005"><div class="list-item-paragraph"><p>find $x$;</p></div></li><li data-list-item-id="ibsl-q033-part-2-2-006" data-list-item-path="ibsl-q033-part-2-004/ibsl-q033-part-2-2-006"><div class="list-item-paragraph">find $P(A ∩ B)$.</div></li></ol></li><li data-list-item-id="ibsl-q033-part-3-007" data-list-item-path="ibsl-q033-part-3-007"><div class="list-item-paragraph"><p>Hence, explain why $A$ and $B$ are not mutually exclusive.</p><span class="inline-marks">[1]</span></div></li></ol>

Let $A$ and $B$ be independent events, where $P(A) = 0.6$ and $P(B) = x$.

Write down an expression for $P(A ∩ B)$.

Mutually Exclusive Events

Use independence to solve an unknown event probability and distinguish mutual exclusivity.

<p>Consider the events A and B, where&nbsp;$P(A) = 0.5$, $P(B) = 0.7$ and $P(A ∩ B) = 0.3$. The Venn diagram below shows the events $A$ and $B$, and the probabilities $p, q$ and $r$.</p>
<div class="graph-block" data-graph-config="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" data-width="520" data-height="260" style="width:520px;height:260px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="abc-list"><li data-list-item-id="ibsl-q009-part-1-014" data-list-item-path="ibsl-q009-part-1-014"><div class="list-item-paragraph">Write down the value of<span class="inline-marks">[3]</span></div><ol class="i-list"><li data-list-item-id="ibsl-q009-part-1-1-015" data-list-item-path="ibsl-q009-part-1-014/ibsl-q009-part-1-1-015"><div class="list-item-paragraph"><p>$p$;</p></div></li><li data-list-item-id="ibsl-q009-part-1-2-016" data-list-item-path="ibsl-q009-part-1-014/ibsl-q009-part-1-2-016"><div class="list-item-paragraph"><p>$q$;</p></div></li><li data-list-item-id="ibsl-q009-part-1-3-017" data-list-item-path="ibsl-q009-part-1-014/ibsl-q009-part-1-3-017"><div class="list-item-paragraph"><p>$r$.</p></div></li></ol></li><li data-list-item-id="ibsl-q009-part-2-018" data-list-item-path="ibsl-q009-part-2-018"><div class="list-item-paragraph"><p>Find the value of $P(A | B')$.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="ibsl-q009-part-3-019" data-list-item-path="ibsl-q009-part-3-019"><div class="list-item-paragraph"><p>Hence, or otherwise, show that the events $A$ and $B$ are not independent.</p><span class="inline-marks">[1]</span></div></li></ol>

Consider the events A and B, where $P(A) = 0.5$, $P(B) = 0.7$ and $P(A ∩ B) = 0.3$. The Venn diagram below shows the events $A$ and $B$, and the probabilities $p, q$ and $r$.

Complementary Events

Complete Venn-region probabilities, calculate a conditional complement and test independence.

<p>The events $A$ and $B$ satisfy $P(A)=\dfrac{2}{5}$, $P(B')=\dfrac{3}{4}$ and $P(B\mid A)=\dfrac{1}{4}$.</p>
<ol class="i-list"><li data-list-item-id="prob-b-item-i" data-list-item-path="prob-b-item-i"><div class="list-item-paragraph">Find $P(A'\cap B')$ and hence prove that the events $A'$ and $B'$ are independent.<span class="inline-marks">[4]</span></div></li></ol>
<p>A third event $C$ satisfies $P(C)=\dfrac{1}{2}$ and $P(A\cap B\cap C)=\dfrac{1}{20}$. It is also known that&nbsp;$B$ and $C$ are independent, while $A$ and $C$ are not independent.</p>
<ol class="i-list" start="2"><li data-list-item-id="prob-b-item-ii" data-list-item-path="prob-b-item-ii"><div class="list-item-paragraph">Find the range of possible values of $P(A'\cap B'\cap C')$.<span class="inline-marks">[5]</span></div></li></ol>

The events $A$ and $B$ satisfy $P(A)=\dfrac{2}{5}$, $P(B')=\dfrac{3}{4}$ and $P(B\mid A)=\dfrac{1}{4}$.

A third event $C$ satisfies $P(C)=\dfrac{1}{2}$ and $P(A\cap B\cap C)=\dfrac{1}{20}$. It is also known that $B$ and $C$ are independent, while $A$ and $C$ are not independent.

Applying Probability Laws

Probability Bounds and Inequalities

Addition Rule for Overlapping Events

<p>For events $A$ and $B$, it is given that&nbsp;$P(A)=\dfrac{1}{4}$, $P(A\cup B)=\dfrac{5}{8}$ and $P(A\cap B')=\dfrac{1}{8}$.</p>
<ol class="i-list"><li data-list-item-id="prob-a-item-i" data-list-item-path="prob-a-item-i"><div class="list-item-paragraph">Find $P(B)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="prob-a-item-ii" data-list-item-path="prob-a-item-ii"><div class="list-item-paragraph">Determine, with a reason, whether the events $A$ and $B$ are independent.<span class="inline-marks">[2]</span></div></li></ol>
<p>For a third event $C$, it is given that&nbsp;$P(C)=\dfrac{2}{5}$, $P(A\cap B\cap C)=\dfrac{1}{20}$, and $C$ is independent of $A$.</p>
<ol class="i-list" start="3"><li data-list-item-id="prob-a-item-iii" data-list-item-path="prob-a-item-iii"><div class="list-item-paragraph">Find $P(A'\cap C)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="prob-a-item-iv" data-list-item-path="prob-a-item-iv"><div class="list-item-paragraph">Find the range of possible values of $P(A'\cap B'\cap C')$.<span class="inline-marks">[3]</span></div></li></ol>

For events $A$ and $B$, it is given that $P(A)=\dfrac{1}{4}$, $P(A\cup B)=\dfrac{5}{8}$ and $P(A\cap B')=\dfrac{1}{8}$.

For a third event $C$, it is given that $P(C)=\dfrac{2}{5}$, $P(A\cap B\cap C)=\dfrac{1}{20}$, and $C$ is independent of $A$.

<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.

Probabilities of Successive Events

Finding Unknowns in Probability Models

Probability Using Permutations and Combinations

Waiting-Time and Stopping-Rule Probabilities

2013 HCI P2 Q10

<p>A box contains $5$ red balls, $5$ green balls and $5$ blue balls. The balls are identical in size. Balls of the same colour are each printed with a distinct number from $1$, $2$, $3$, $4$, $5$. Two balls are randomly drawn without replacement. Find the probability that</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">at least $1$ red ball is drawn,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the sum of the numbers on the balls drawn is at least $9$,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the sum of the numbers on the balls drawn is at least $9$ given that at least $1$ red ball is drawn.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>State, with a reason, whether the events 'the sum of the numbers on the balls drawn is at least $9$' and 'at least $1$ red ball is drawn' are independent.<span class="inline-marks">[1]</span></p>

A box contains $5$ red balls, $5$ green balls and $5$ blue balls. The balls are identical in size. Balls of the same colour are each printed with a distinct number from $1$, $2$, $3$, $4$, $5$. Two balls are randomly drawn without replacement. Find the probability that

the sum of the numbers on the balls drawn is at least $9$,

the sum of the numbers on the balls drawn is at least $9$ given that at least $1$ red ball is drawn.

State, with a reason, whether the events 'the sum of the numbers on the balls drawn is at least $9$' and 'at least $1$ red ball is drawn' are independent.

Selections with Restrictions

Complementary Counting

2013 DHS P2 Q7

Permutation and Combination

<p>A bag contains $4$ red discs, $10$ white discs and $6$ blue discs. Two discs are selected at random from the bag, one at a time with replacement.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Draw a tree diagram to illustrate the experiment.</span></div></li><li><div class="list-item-paragraph">Using the tree diagram in <strong>(a)</strong>, find the probability of selecting two discs of different colours.</div></li></ol>

A bag contains $4$ red discs, $10$ white discs and $6$ blue discs. Two discs are selected at random from the bag, one at a time with replacement.

Draw a tree diagram to illustrate the experiment.

Constructing and Using Probability Trees

Sampling with Replacement

<p>Susan has&nbsp;$2$ white and $6$ blue blouses, and $10$ skirts of which $6$ are white and $4$ are yellow, in her cupboard. Every morning, she selects a blouse and a skirt at random.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Copy and complete the tree diagram below.</span></div></li></ol>
<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/1767701003751-qi3mssv65h.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3MDEwMDM3NTEtcWkzbXNzdjY1aC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTY2NTM1LCJleHAiOjE3OTM1NjY4MzV9.e9SMQolVDfHzt7LkOh-mpSNcFy8wRU1sNfqix-Sv5BU" style="max-width: min(300px, 100%); height: auto; display: block;" alt="Tree diagram for selecting a blouse and a skirt." class="aligncenter custom-width" /></figure></div>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Calculate the probability that Susan</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">selects a blouse and a skirt of the same colour,</span></div></li><li><div class="list-item-content"><span class="list-item-text">does not select a blue blouse and a yellow skirt.</span></div></li></ol></li></ol>

Susan has $2$ white and $6$ blue blouses, and $10$ skirts of which $6$ are white and $4$ are yellow, in her cupboard. Every morning, she selects a blouse and a skirt at random.

Copy and complete the tree diagram below.

Probability Trees for Clothing Choices

<p>A dart board made up of $3$ circles with $3$ different colours is shown in the diagram. Each circle has a radius of $2$ cm, $6$ cm and $8$ cm respectively. Assume that all darts thrown hit the board.</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 data-list-item-id="item-1773731735577-c4upj30qs" data-list-item-path="item-1773731735577-c4upj30qs"><div class="list-item-paragraph">Show that the probability of a dart hitting the red area is $\frac{1}{16}$.</div></li><li data-list-item-id="item-1773731736094-cpv90njoh" data-list-item-path="item-1773731736094-cpv90njoh"><div class="list-item-paragraph">If&nbsp;$2$ darts are thrown, one at a time, find the probability of both darts hitting the orange area.</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/1767699347486-1b2bgfgqzq3.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTkzNDc0ODYtMWIyYmdmZ3F6cTMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU2NjUzNSwiZXhwIjoxNzkzNTY2ODM1fQ.zRDpg1VVm1U86dMnyM3hpnaOUFs9ssAbgM5DyP1j0w8" style="max-width: min(240px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>
<p>Assume each dart lands uniformly over the area of the board, and the two throws are independent.</p>

A dart board made up of $3$ circles with $3$ different colours is shown in the diagram. Each circle has a radius of $2$ cm, $6$ cm and $8$ cm respectively. Assume that all darts thrown hit the board.

Assume each dart lands uniformly over the area of the board, and the two throws are independent.

Probability of a Single Event

Circumference and Area of Circles

Geometric Probability on a Dartboard

Perimeter and Area of Plane Figures

<p>Patricia throws a fair $6$-sided die and a fair coin together.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Draw a tree diagram to display all the possible outcomes.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability of obtaining</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">an odd number and a head,</span></div></li><li><div class="list-item-content"><span class="list-item-text">a prime number and a tail,</span></div></li><li><div class="list-item-content"><span class="list-item-text">a ‘$6$’ and a tail.</span></div></li></ol></li></ol>

Patricia throws a fair $6$-sided die and a fair coin together.

Draw a tree diagram to display all the possible outcomes.

Constructing Probability Trees for Multi-Variable Sets

<p>A box contains $3$ balls labelled $3$, $6$ and $9$. A second box contains another $3$ balls labelled $4$, $6$ and $8$. A ball is drawn at random from each box and the number is recorded.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Draw a tree diagram to show 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 sum of the two numbers is more than $10$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the product of the two numbers is a perfect square.</span></div></li></ol></li></ol>

A box contains $3$ balls labelled $3$, $6$ and $9$. A second box contains another $3$ balls labelled $4$, $6$ and $8$. A ball is drawn at random from each box and the number is recorded.

Draw a tree diagram to show all the possible outcomes.

Calculations of Dependent vs. Independent Event Trials

<p>Roger throws a fair coin and a fair $5$-sided die, with sides numbered $1$ to $5$, at the same time.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Draw a tree diagram to display all the possible outcomes.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability of getting</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">a tail and an even number,</span></div></li><li><div class="list-item-content"><span class="list-item-text">a tail and an odd number,</span></div></li><li><div class="list-item-content"><span class="list-item-text">a head and the number ‘$4$’.</span></div></li></ol></li></ol>

Roger throws a fair coin and a fair $5$-sided die, with sides numbered $1$ to $5$, at the same time.

Discrete Sample Spaces and Outcome Tables

<p>A box contains $2$ identical pink marbles and $6$ identical blue marbles. A marble was taken out at random from the box, the colour was noted and the marble was replaced. A second marble was then taken from the box. Find the probability that the marbles were</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">both blue,</span></div></li><li><div class="list-item-content"><span class="list-item-text">of different colours.</span></div></li></ol>

A box contains $2$ identical pink marbles and $6$ identical blue marbles. A marble was taken out at random from the box, the colour was noted and the marble was replaced. A second marble was then taken from the box. Find the probability that the marbles were

Probability of Independent Events: Modeling Multi-Stage Experiments

<p>Three fair $6$-sided dice are thrown at the same time, what is the probability that all the three dice</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">show even numbers?</span></div></li><li><div class="list-item-content"><span class="list-item-text">show different numbers?</span></div></li></ol>

Three fair $6$-sided dice are thrown at the same time, what is the probability that all the three dice

Probability of Independent Events: Simultaneous Dice Trials

<p>A pizza shop sells pizzas with a thin or thick crust together with one of three different toppings. The three different toppings are mushroom, onions and bacon. Kenny buys a pizza from the pizza shop. He chooses each crust with equal probability and independently chooses each of the three toppings with equal probability.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Display all the possible pizza combinations using a tree diagram.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the probability that the pizza Kenny buys</div><ol class="i-list"><li data-list-item-id="item-b-i" data-list-item-path="item-b/item-b-i"><div class="list-item-paragraph">does not have onions,</div></li><li data-list-item-id="item-b-ii" data-list-item-path="item-b/item-b-ii"><div class="list-item-paragraph">does not have mushroom or onions.</div></li></ol></li></ol>

A pizza shop sells pizzas with a thin or thick crust together with one of three different toppings. The three different toppings are mushroom, onions and bacon. Kenny buys a pizza from the pizza shop. He chooses each crust with equal probability and independently chooses each of the three toppings with equal probability.

Display all the possible pizza combinations using a tree diagram.

Find the probability that the pizza Kenny buys

Sequential Probability: Construction and Analysis of Tree Diagrams

<p>A spinner with equal sectors is shown. The spinner is spun once and the number of the sector the spinner lands on is observed. Each time the spinner is spun, it is equally likely to stop at any sector.<br>At the same time the spinner is spun, a fair coin is also tossed once.</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">Represent the possible outcomes using a tree diagram.</div></li><li><div class="list-item-paragraph">Calculate the probability that</div><ol class="i-list"><li><div class="list-item-paragraph">the spinner lands on an even number and the coin shows head,</div></li><li><div class="list-item-paragraph">the spinner lands on an odd number and the coin shows tail.</div></li></ol></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/1766142915189-vlulrv21swr.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjYxNDI5MTUxODktdmx1bHJ2MjFzd3IucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU2NjUzNSwiZXhwIjoxNzkzNTY2ODM1fQ.Cx-dnElDpLCm8K5ZcNLFfN0i68-KsG4ZLDwujQN1ulA" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

A spinner with equal sectors is shown. The spinner is spun once and the number of the sector the spinner lands on is observed. Each time the spinner is spun, it is equally likely to stop at any sector.
At the same time the spinner is spun, a fair coin is also tossed once.

Mastery of Tree Diagrams: Calculating Joint Probabilities

<p>A box contains $9$ red cards, $6$ blue cards and $3$ yellow cards.</p>
<p>Two cards are drawn at random, without replacement, from the box.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Draw a tree diagram to show the probabilities of the possible outcomes.</div></li><li><div class="list-item-paragraph">Find, as a fraction in its simplest form, the probability that</div><ol class="i-list"><li><div class="list-item-paragraph">the two cards are of the same colour,</div></li><li><div class="list-item-paragraph">the two cards are of different colours.</div></li></ol></li><li><div class="list-item-paragraph">All the cards are placed back in the box.</div><div class="list-item-paragraph">The cards are then drawn at random, with replacement, from the box.</div><ol class="i-list"><li><div class="list-item-paragraph">Find the probability that exactly $2$ blue cards are being drawn out of $3$ draws.</div></li><li><div class="list-item-paragraph">Find the probability that at least a yellow card is being drawn out of $5$ draws.</div></li></ol></li></ol>

A box contains $9$ red cards, $6$ blue cards and $3$ yellow cards.

Two cards are drawn at random, without replacement, from the box.

2020 HCI EM S4 P2

<p>For events $A$ and $B$, it is given that&nbsp;$P\left( A \right)=\frac{1}{8}$, $P\left( A\cup B \right)=\frac{9}{16}$ and $P\left( A\cap {B}' \right)=\frac{1}{16}$.</p>
<ol class="i-list"><li data-list-item-id="item-1760781172283-qq04lwcra" data-list-item-path="item-1760781172283-qq04lwcra"><div class="list-item-paragraph">Find $P\left( B \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760781174061-kzrl5fl7k" data-list-item-path="item-1760781174061-kzrl5fl7k"><div class="list-item-paragraph">Determine, with a reason, whether the events $A$ and $B$ are independent.<span class="inline-marks">[2]</span></div></li></ol>
<p>For a third event $C$, it is given that&nbsp;$P\left( C \right)=\frac{3}{8}$, $P\left( A\cap B\cap C \right)=\frac{1}{64}$ and $C$ is independent of $A$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760781194927-07nr12r5q" data-list-item-path="item-1760781194927-07nr12r5q"><div class="list-item-paragraph">Find $P\left( {A}'\cap C \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760781195819-twm664oux" data-list-item-path="item-1760781195819-twm664oux"><div class="list-item-paragraph">Find the range of values for&nbsp;$P\left( A'\cap {B}'\cap {C}' \right)$.<span class="inline-marks">[3]</span></div></li></ol>

For events $A$ and $B$, it is given that $P\left( A \right)=\frac{1}{8}$, $P\left( A\cup B \right)=\frac{9}{16}$ and $P\left( A\cap {B}' \right)=\frac{1}{16}$.

For a third event $C$, it is given that $P\left( C \right)=\frac{3}{8}$, $P\left( A\cap B\cap C \right)=\frac{1}{64}$ and $C$ is independent of $A$.

2024 TMJC P2 Q5

<p>The events $A$ and $B$ are such that&nbsp;$\mathrm{P}\left( A \right)=a$ and $\mathrm{P}\left( B \right)=b$. $A$ and $B$ are independent events.</p>
<ol class="abc-list"><li data-list-item-id="item-1760781012593-etiznm953" data-list-item-path="item-1760781012593-etiznm953"><div class="list-item-paragraph">Find an expression for&nbsp;$\mathrm{P}\left( {A}'\cap {B}' \right)$ in terms of $a$ and $b$, and hence prove that&nbsp;${A}'$ and ${B}'$ are independent events.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is given that&nbsp;$\mathrm{P}\left( A\mid {B}' \right)=0.85$ and ${\mathrm{P}\left( {{B}'} \right)=0.8}$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760781026636-y4nwlf8ee" data-list-item-path="item-1760781026636-y4nwlf8ee"><div class="list-item-paragraph">Find $\mathrm{P}\left( A\cap {B}' \right)$.<span class="inline-marks">[2]</span></div></li></ol>
<p>For a third event $C$, it is given that&nbsp;$A$ and $C$ are mutually exclusive and $\mathrm{P}\left( {A}'\cap {C}' \right)=0.52$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-1760781041671-n28p3vsy0" data-list-item-path="item-1760781041671-n28p3vsy0"><div class="list-item-paragraph">Find $\mathrm{P}\left( C \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760781044055-x873mmx49" data-list-item-path="item-1760781044055-x873mmx49"><div class="list-item-paragraph">Hence find the set of possible values of $\mathrm{P}\left( {A}'\cap {B}'\cap {C}' \right)$.<span class="inline-marks">[3]</span></div></li></ol>

The events $A$ and $B$ are such that $\mathrm{P}\left( A \right)=a$ and $\mathrm{P}\left( B \right)=b$. $A$ and $B$ are independent events.

It is given that $\mathrm{P}\left( A\mid {B}' \right)=0.85$ and ${\mathrm{P}\left( {{B}'} \right)=0.8}$.

For a third event $C$, it is given that $A$ and $C$ are mutually exclusive and $\mathrm{P}\left( {A}'\cap {C}' \right)=0.52$.

Math Question Bank