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 diagram below shows the probabilities for events $A$ and $B$, with $P(A') = p$.</p>
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    <foreignObject x="184.1213743357479" y="222.044849502201" width="58" height="30">
      <div xmlns="http://www.w3.org/1999/xhtml" class="flowchart-edge-label probability-tree-outcome-label" style="box-sizing:border-box;width:100%;height:100%;display:flex;align-items:center;justify-content:center;overflow:visible;padding:2px 5px;border-radius:4px;background-color:transparent;color:#111827;font-family:Arial, sans-serif;font-size:12px;line-height:1.15;text-align:center;white-space:pre-wrap;overflow-wrap:anywhere">$A&#39;$</div>
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      <path d="M 255,82L 455,30" fill="none" stroke="#111827" stroke-width="1.6" stroke-linecap="round" stroke-linejoin="round" />
      
    <foreignObject x="290.9771260752907" y="27.450484904964192" width="121" height="30">
      <div xmlns="http://www.w3.org/1999/xhtml" class="flowchart-edge-label probability-tree-probability-label" style="box-sizing:border-box;width:100%;height:100%;display:flex;align-items:center;justify-content:center;overflow:visible;padding:2px 5px;border-radius:4px;background-color:transparent;color:#111827;font-family:Arial, sans-serif;font-size:12px;line-height:1.15;text-align:center;white-space:pre-wrap;overflow-wrap:anywhere">$\frac{1}{4}$</div>
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    <foreignObject x="467" y="15" width="51" height="30">
      <div xmlns="http://www.w3.org/1999/xhtml" class="flowchart-edge-label probability-tree-outcome-label" style="box-sizing:border-box;width:100%;height:100%;display:flex;align-items:center;justify-content:flex-start;overflow:visible;padding:2px 5px;border-radius:4px;background-color:transparent;color:#111827;font-family:Arial, sans-serif;font-size:12px;line-height:1.15;text-align:center;white-space:pre-wrap;overflow-wrap:anywhere">$B$</div>
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      <path d="M 255,82L 455,110" fill="none" stroke="#111827" stroke-width="1.6" stroke-linecap="round" stroke-linejoin="round" />
      
    <foreignObject x="292.5589301765183" y="94.86478445344062" width="121" height="30">
      <div xmlns="http://www.w3.org/1999/xhtml" class="flowchart-edge-label probability-tree-probability-label" style="box-sizing:border-box;width:100%;height:100%;display:flex;align-items:center;justify-content:center;overflow:visible;padding:2px 5px;border-radius:4px;background-color:transparent;color:#111827;font-family:Arial, sans-serif;font-size:12px;line-height:1.15;text-align:center;white-space:pre-wrap;overflow-wrap:anywhere">$\frac{3}{4}$</div>
    </foreignObject>
  
      
    <foreignObject x="467" y="95" width="58" height="30">
      <div xmlns="http://www.w3.org/1999/xhtml" class="flowchart-edge-label probability-tree-outcome-label" style="box-sizing:border-box;width:100%;height:100%;display:flex;align-items:center;justify-content:flex-start;overflow:visible;padding:2px 5px;border-radius:4px;background-color:transparent;color:#111827;font-family:Arial, sans-serif;font-size:12px;line-height:1.15;text-align:center;white-space:pre-wrap;overflow-wrap:anywhere">$B&#39;$</div>
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    <foreignObject x="292.5589301765183" y="195.13521554655938" width="121" height="30">
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    <foreignObject x="467" y="195" width="51" height="30">
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    </foreignObject>
  
      
    
      <path d="M 255,238L 455,290" fill="none" stroke="#111827" stroke-width="1.6" stroke-linecap="round" stroke-linejoin="round" />
      
    <foreignObject x="290.9771260752907" y="262.5495150950358" width="121" height="30">
      <div xmlns="http://www.w3.org/1999/xhtml" class="flowchart-edge-label probability-tree-probability-label" style="box-sizing:border-box;width:100%;height:100%;display:flex;align-items:center;justify-content:center;overflow:visible;padding:2px 5px;border-radius:4px;background-color:transparent;color:#111827;font-family:Arial, sans-serif;font-size:12px;line-height:1.15;text-align:center;white-space:pre-wrap;overflow-wrap:anywhere">$\frac{5}{8}$</div>
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    <foreignObject x="467" y="275" width="58" height="30">
      <div xmlns="http://www.w3.org/1999/xhtml" class="flowchart-edge-label probability-tree-outcome-label" style="box-sizing:border-box;width:100%;height:100%;display:flex;align-items:center;justify-content:flex-start;overflow:visible;padding:2px 5px;border-radius:4px;background-color:transparent;color:#111827;font-family:Arial, sans-serif;font-size:12px;line-height:1.15;text-align:center;white-space:pre-wrap;overflow-wrap:anywhere">$B&#39;$</div>
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  </div>
<ol class="abc-list"><li data-list-item-id="ibsl-q014-part-1-017" data-list-item-path="ibsl-q014-part-1-017"><div class="list-item-paragraph"><p>Write down the value of $p$.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="ibsl-q014-part-2-018" data-list-item-path="ibsl-q014-part-2-018"><div class="list-item-paragraph">Find $P(B)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="ibsl-q014-part-3-019" data-list-item-path="ibsl-q014-part-3-019"><div class="list-item-paragraph"><p>Find $P(A' | B)$.</p><span class="inline-marks">[3]</span></div></li></ol>

The diagram below shows the probabilities for events $A$ and $B$, with $P(A') = p$.

Dependent Events

Constructing and Using Probability Trees

Read a probability tree and calculate a conditional probability.

Probability

<p></p>
<p>A bag contains $4$ black balls and $6$ white balls. Three balls are drawn from the bag at random, one by one. If a black ball is drawn, it is replaced in the bag. If a white ball is drawn, it is not replaced and $2$ extra black balls are added to the bag.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that the probability that all balls drawn are white is $\frac{1}{11}$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that at least one of the balls drawn is black, find the probability that all balls drawn are black.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>A fourth ball is further drawn from the bag at random.</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the probability that the third white ball appears on this fourth draw.</span><span class="inline-marks">[3]</span></div></li></ol>

A bag contains $4$ black balls and $6$ white balls. Three balls are drawn from the bag at random, one by one. If a black ball is drawn, it is replaced in the bag. If a white ball is drawn, it is not replaced and $2$ extra black balls are added to the bag.

Show that the probability that all balls drawn are white is $\frac{1}{11}$.

Given that at least one of the balls drawn is black, find the probability that all balls drawn are black.

A fourth ball is further drawn from the bag at random.

Find the probability that the third white ball appears on this fourth draw.

Conditional Probability

Probabilities of Successive Events

Conditional Probability from Tree Diagrams

Waiting-Time and Stopping-Rule Probabilities

2013 JJC P2 Q9

<p>In a box, there are $12$ marbles of which $5$ are red, $4$ are black and $3$ are white. $4$ marbles are taken at random from the box, each marble being replaced before the next one is taken. Find the probability that</p>
<ol class="abc-list"><li><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">at least one marble of each colour is obtained,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">at least one marble is white.</span><span class="inline-marks">[2]</span></div></li></ol></li></ol>
<p>Everyday, Gerald has a choice of $2$ routes to get to school. The probability that he gets to school without being delayed is $\frac{43}{50}$. The probability that he chooses the first route is $\frac{3}{5}$. If he gets to school without being delayed, the probability that the first route is chosen is $\frac{27}{43}$. Find the probability that he gets to school without being delayed if</p>
<ol class="abc-list" start="2"><li><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">he chooses the first route,</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">he chooses the second route.</span><span class="inline-marks">[3]</span></div></li></ol></li></ol>

In a box, there are $12$ marbles of which $5$ are red, $4$ are black and $3$ are white. $4$ marbles are taken at random from the box, each marble being replaced before the next one is taken. Find the probability that

Everyday, Gerald has a choice of $2$ routes to get to school. The probability that he gets to school without being delayed is $\frac{43}{50}$. The probability that he chooses the first route is $\frac{3}{5}$. If he gets to school without being delayed, the probability that the first route is chosen is $\frac{27}{43}$. Find the probability that he gets to school without being delayed if

Bayes' Theorem

Complementary Events

Probability Using Permutations and Combinations

2013 TJC P2 Q10

<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.

Independent Events

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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3MDEwMDM3NTEtcWkzbXNzdjY1aC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzQ5NzQ1LCJleHAiOjE3OTI3NTAwNDV9.AQl2aBUOdNdqDDykasgafP7P09PAnIyB2LU1lv0WsGQ" 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 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTg3MDE2NjQtMXR2YnFhMWg0NXcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTc0NSwiZXhwIjoxNzkyNzUwMDQ1fQ.wWdFvdB3M8tXVjafdwYpzAh-2Ku--58wOSUYn2pdyP0" 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

Finding Unknowns in Probability Models

Theoretical Probability in Successive Experiments

<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 container holds roses of three colours. There are $4$ pink, $5$ blue and $6$ red roses. Linda picks $2$ roses at random, one after the other without 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-content"><span class="list-item-text">Calculate the probability that</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">both roses are of the same colour,</span></div></li><li><div class="list-item-content"><span class="list-item-text">one of the roses is red and the other rose is blue.</span></div></li></ol></li></ol>

A container holds roses of three colours. There are $4$ pink, $5$ blue and $6$ red roses. Linda picks $2$ roses at random, one after the other without replacement.

Application of Tree Diagrams in Sampling

<p>In a class of $25$ students, $14$ are boys and $11$ are girls. A teacher selects a student at random from the class. Another student is selected at random from the remaining students.</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">Copy and complete the tree diagram below.</div></li><li><div class="list-item-paragraph">Find the probability that</div><ol class="i-list"><li><div class="list-item-paragraph">the first student selected is a girl, and the second student is a boy,</div></li><li><div class="list-item-paragraph">only one of the two students selected is a girl,</div></li><li><div class="list-item-paragraph">the second student selected is a boy.</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/1766144163628-ajcq1ng243l.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjYxNDQxNjM2MjgtYWpjcTFuZzI0M2wucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTc0NSwiZXhwIjoxNzkyNzUwMDQ1fQ.yzx62cBZpPYAJ2kcLoG9QL9GJb973yvcPf0iSY84P-8" style="max-width: min(250px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

In a class of $25$ students, $14$ are boys and $11$ are girls. A teacher selects a student at random from the class. Another student is selected at random from the remaining students.

Law of Total Probability

Probability Trees for Dependent Events

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjYxNDI5MTUxODktdmx1bHJ2MjFzd3IucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTc0NSwiZXhwIjoxNzkyNzUwMDQ1fQ.KdXev7UPtS-V-8SEi7CfKK_fBZ84i5yy1cnYkhOOQTM" 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>In a game of chance, a player has to spin a fair spinner. The spinner has $7$ sections and an arrow which has an equal chance of coming to rest over any of the $7$ sections. The spinner has $1$ section labelled $\mathbf{R}$, $2$ sections labelled $\mathbf{B}$ and $4$ sections labelled $\mathbf{Y}$ (see diagram).</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760955584642-1qpwfpsbvyoi.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA5NTU1ODQ2NDItMXFwd2Zwc2J2eW9pLnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTI3NDk3NDUsImV4cCI6MTc5Mjc1MDA0NX0.yd8XgxjBh3Y50bcKY2l6Vl1xpo8ovsM_ZuxwrlY-Blo" width="400" class="aligncenter" /></figure>
<p>The player then has to throw one of three fair six-sided dice, coloured red, blue or yellow. If the spinner comes to rest over $\mathbf{R}$ the red die is thrown, if the spinner comes to rest over $\mathbf{B}$ the blue die is thrown and if the spinner comes to rest over $\mathbf{Y}$ the yellow die is thrown. The yellow die has one face with $*$ on it, the blue die has two faces with $*$ on it and the red die has three faces with $*$ on it. The player wins the game if the die thrown comes to rest with a face showing $*$ uppermost.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the probability that a player wins a game.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Given that a player wins a game, find the probability that the spinner came to rest over $B$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the probability that a player wins $3$ consecutive games, each time throwing a die of a different colour.</div><p style="text-align: right;">[2]</p></li></ol>

In a game of chance, a player has to spin a fair spinner. The spinner has $7$ sections and an arrow which has an equal chance of coming to rest over any of the $7$ sections. The spinner has $1$ section labelled $\mathbf{R}$, $2$ sections labelled $\mathbf{B}$ and $4$ sections labelled $\mathbf{Y}$ (see diagram).

The player then has to throw one of three fair six-sided dice, coloured red, blue or yellow. If the spinner comes to rest over $\mathbf{R}$ the red die is thrown, if the spinner comes to rest over $\mathbf{B}$ the blue die is thrown and if the spinner comes to rest over $\mathbf{Y}$ the yellow die is thrown. The yellow die has one face with $*$ on it, the blue die has two faces with $*$ on it and the red die has three faces with $*$ on it. The player wins the game if the die thrown comes to rest with a face showing $*$ uppermost.

N2016 P2 Q5

<p>A dart player, Mike, in Smart Andrew’s College Darts Team is having a practice session of dart throwing in the hall. The probability that Mike hits his target on the first throw is $0.7$.</p>
<p>For each throw after his first throw, the probability that Mike hits the target, given that he hit the target on his preceding throw, is $0.77$.</p>
<p>For each throw after his first throw, the probability that Mike does not hit the target, given that he did not hit the target on his preceding throw, is $0.33$.</p>
<ol class="i-list"><li data-list-item-id="item-1760783192258-ct6klzlos" data-list-item-path="item-1760783192258-ct6klzlos"><div class="list-item-paragraph">Draw a fully labelled tree diagram to represent Mike’s first two throws.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760783192993-31duwnd34" data-list-item-path="item-1760783192993-31duwnd34"><div class="list-item-paragraph">Find the probability that Mike hits his target on the second throw of the dart.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is now known that Mike threw the dart three times.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760783204528-f16jotcy9" data-list-item-path="item-1760783204528-f16jotcy9"><div class="list-item-paragraph">Given that Mike did not hit his target on the second throw, find the probability that he hit the target twice.<span class="inline-marks">[3]</span></div></li></ol>

A dart player, Mike, in Smart Andrew’s College Darts Team is having a practice session of dart throwing in the hall. The probability that Mike hits his target on the first throw is $0.7$.

For each throw after his first throw, the probability that Mike hits the target, given that he hit the target on his preceding throw, is $0.77$.

For each throw after his first throw, the probability that Mike does not hit the target, given that he did not hit the target on his preceding throw, is $0.33$.

It is now known that Mike threw the dart three times.

2024 SAJC P2 Q5

<p>A company sells consumer hobby drones, and they buy the rotor motor parts from two suppliers, AE Rotors and Brushless Inc. ${25\%}$ of the motors are from AE Rotors and the rest are from Brushless Inc. The probability that a randomly chosen motor supplied by AE Rotors is faulty is $\frac{1}{20}$. The probability that a randomly chosen motor supplied by Brushless Inc is faulty is $q$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760781269010-ow7oymcdt" data-list-item-path="item-1760781269010-ow7oymcdt"><div class="list-item-paragraph">Complete the tree diagram below to illustrate the possibilities for a randomly chosen motor. You should use the following notations for the events:</div><div class="list-item-paragraph" style="text-align: center"><p style="text-align: left;">$A$ for the motor is supplied by AE Rotors. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $B$ for the motor is supplied by Brushless Inc.</p></div><div class="list-item-paragraph" style="text-align: center"><p style="text-align: left;">$F$ for the motor is faulty, and &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $\quad\,{F}'$ for its complement, i.e. the motor is not faulty.</p><span class="inline-marks">[1]</span></div></li></ol>
<p>It is known that the probability that a randomly picked motor that is faulty is supplied by Brushless Inc is $\frac{3}{8}$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760781281010-gszqiiyg5" data-list-item-path="item-1760781281010-gszqiiyg5"><div class="list-item-paragraph">Use the above information to set up an equation involving $q$, and hence find the value of $q$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760781281843-6x9qpzzhr" data-list-item-path="item-1760781281843-6x9qpzzhr"><div class="list-item-paragraph">Three motors are randomly picked. Find the probability that all of them are supplied by AE Rotors and exactly one of them is faulty.<span class="inline-marks">[2]</span></div></li></ol>

A company sells consumer hobby drones, and they buy the rotor motor parts from two suppliers, AE Rotors and Brushless Inc. ${25\%}$ of the motors are from AE Rotors and the rest are from Brushless Inc. The probability that a randomly chosen motor supplied by AE Rotors is faulty is $\frac{1}{20}$. The probability that a randomly chosen motor supplied by Brushless Inc is faulty is $q$.

It is known that the probability that a randomly picked motor that is faulty is supplied by Brushless Inc is $\frac{3}{8}$.

Binomial Probabilities Using the Probability Mass Function

2024 EJC P2 Q5

Discrete Random Variables (DRV)

<p>$16\%$ of all phone calls are scam calls. A company designed a $2$-stage filtering anti-scam app, ${{X}^{2}}$scam, that can block scam calls which plague mobile phone users. In a pre-launch test, it is found that ${{X}^{2}}$scam blocked $88\%$ of calls that are scam calls and $5\%$ of calls that are non-scam calls in the first stage of filtering. Of the calls that are unblocked in the first stage, ${{X}^{2}}$scam blocks $99\%$of scam calls and $5\%$ of non-scam calls within the second stage. Calls that are blocked in the first stage do not go through a second stage of filtering.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Draw a tree diagram to represent the above information.</div><p style="text-align: right;">[2]</p></li></ol>
<p>A call is selected at random. Find the probability that</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">it is a scam call that is blocked.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">it is a scam call, given that the call is blocked.</div><p style="text-align: right;">[2]</p></li></ol>
<p>Three calls were selected at random.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find the probability that ${{X}^{2}}$scam correctly filters exactly two of the calls.</div><p style="text-align: right;">[4]</p></li></ol>

$16\%$ of all phone calls are scam calls. A company designed a $2$-stage filtering anti-scam app, ${{X}^{2}}$scam, that can block scam calls which plague mobile phone users. In a pre-launch test, it is found that ${{X}^{2}}$scam blocked $88\%$ of calls that are scam calls and $5\%$ of calls that are non-scam calls in the first stage of filtering. Of the calls that are unblocked in the first stage, ${{X}^{2}}$scam blocks $99\%$of scam calls and $5\%$ of non-scam calls within the second stage. Calls that are blocked in the first stage do not go through a second stage of filtering.

A call is selected at random. Find the probability that

2022 EJC J2 MYE P2 Q9

<p>It is known that in a certain country, $30\%$ of the people support Party A; $50\%$ of the people support Party B; and $20\%$ of the people support Party C.</p>
<p>In the recent election, $55\%$ of Party A supporters voted, as did $86\%$ of the supporters of Party B and $42\%$ of the supporters of Party C.</p>
<ol class="abc-list"><li data-list-item-id="item-1760434394931-tarcz50ty" data-list-item-path="item-1760434394931-tarcz50ty"><div class="list-item-paragraph">One person is selected at random. Find the probability that</div><ol class="i-list"><li data-list-item-id="subitem-1760434405739-kzhw65rfc" data-list-item-path="item-1760434394931-tarcz50ty/subitem-1760434405739-kzhw65rfc"><div class="list-item-paragraph">the person voted,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760434412008-t3wfs4g6h" data-list-item-path="item-1760434394931-tarcz50ty/subitem-1760434412008-t3wfs4g6h"><div class="list-item-paragraph">the person is a supporter of Party B, given that the person voted.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-1760434395741-gus7rlq94" data-list-item-path="item-1760434395741-gus7rlq94"><div class="list-item-paragraph">Two persons are selected at random. Find the probability that exactly one of them voted.<span class="inline-marks">[2]</span></div></li></ol>

It is known that in a certain country, $30\%$ of the people support Party A; $50\%$ of the people support Party B; and $20\%$ of the people support Party C.

In the recent election, $55\%$ of Party A supporters voted, as did $86\%$ of the supporters of Party B and $42\%$ of the supporters of Party C.

2023 VJC P2 Q6

<p>Diabetes is present in $27\%$ of the elderly population. In a health screening exercise, a quick test for diabetes is used, but the test is not totally reliable. It has&nbsp;$0.98$ probability of giving a positive result when an elderly has diabetes, and $0.04$ probability of giving a positive result when an elderly does not have diabetes. A randomly chosen elderly is given the test. Find the probability that</p>
<ol class="i-list"><li data-list-item-id="item-1760431941810-k5ultf6kk" data-list-item-path="item-1760431941810-k5ultf6kk"><div class="list-item-paragraph">the result of the test is positive,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760431942403-xdfkx3fbm" data-list-item-path="item-1760431942403-xdfkx3fbm"><div class="list-item-paragraph">the elderly has diabetes given that the result of the test is positive.<span class="inline-marks">[2]</span></div></li></ol>

Diabetes is present in $27\%$ of the elderly population. In a health screening exercise, a quick test for diabetes is used, but the test is not totally reliable. It has $0.98$ probability of giving a positive result when an elderly has diabetes, and $0.04$ probability of giving a positive result when an elderly does not have diabetes. A randomly chosen elderly is given the test. Find the probability that

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