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E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

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N(A) A Math

N(A) E Math

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>
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<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>A company uses two machines, $A$ and $B$, to make boxes. Machine $A$ makes $60 \%$ of the boxes. $80 \%$ of the boxes made by machine $A$ pass inspection. $90 \%$ of the boxes made by machine $B$ pass inspection. $A$ box is selected at random.</p>
<ol class="abc-list"><li data-list-item-id="ibsl-q006-part-1-003" data-list-item-path="ibsl-q006-part-1-003"><div class="list-item-paragraph">Find the probability that it passes inspection.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="ibsl-q006-part-2-004" data-list-item-path="ibsl-q006-part-2-004"><div class="list-item-paragraph"><p>The company would like the probability that a box passes inspection to be $0.87$. Find the percentage of boxes that should be made by machine $B$ to achieve this.</p><span class="inline-marks">[4]</span></div></li></ol>

A company uses two machines, $A$ and $B$, to make boxes. Machine $A$ makes $60 \%$ of the boxes. $80 \%$ of the boxes made by machine $A$ pass inspection. $90 \%$ of the boxes made by machine $B$ pass inspection. $A$ box is selected at random.

Find the probability that it passes inspection.

Use total probability for two manufacturing machines and solve an inverse mixture proportion.

<p>The diagram below shows the probabilities for events $A$ and $B$, with $P(A') = p$.</p>
<div class="flowchart-block probability-tree-block" data-width="620" data-height="320" style="width:620px;max-width:100%;display:block;margin:0 auto;">
    <svg
      xmlns="http://www.w3.org/2000/svg"
      width="620"
      height="320"
      viewBox="0 0 620 320"
      role="img"
      aria-label="Probability tree"
    >
      <defs>
        
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      <rect x="0" y="0" width="620" height="320" fill="#ffffff" />
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      <path d="M 50,160L 255,82" fill="none" stroke="#111827" stroke-width="1.6" stroke-linecap="round" stroke-linejoin="round" />
      
    <foreignObject x="87.02137433574791" y="92.91515049779898" 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">$\frac{1}{5}$</div>
    </foreignObject>
  
      
    <foreignObject x="187.6213743357479" y="67.95515049779898" 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: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">$A$</div>
    </foreignObject>
  
      
    
      <path d="M 50,160L 255,238" fill="none" stroke="#111827" stroke-width="1.6" stroke-linecap="round" stroke-linejoin="round" />
      
    <foreignObject x="122.02137433574791" y="197.08484950220102" width="51" 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">$p$</div>
    </foreignObject>
  
      
    <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>
    </foreignObject>
  
      
    
      <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>
    </foreignObject>
  
      
    <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>
    </foreignObject>
  
      
    
      <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>
    </foreignObject>
  
      
    
      <path d="M 255,238L 455,210" fill="none" stroke="#111827" stroke-width="1.6" stroke-linecap="round" stroke-linejoin="round" />
      
    <foreignObject x="292.5589301765183" y="195.13521554655938" 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">$\frac{3}{8}$</div>
    </foreignObject>
  
      
    <foreignObject x="467" y="195" 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>
    </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>
    </foreignObject>
  
      
    <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>
    </foreignObject>
  
      
    
      </g>
      <g class="flowchart-nodes">
        
      </g>
    </svg>
  </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 $P(A') = p$.

Constructing and Using Probability Trees

Read a probability tree and calculate a conditional probability.

<p>The letters of the word $\mathrm{PROBABILITY}$ are written on $11$ cards: $\mathrm{P, R, O, B, A, B, I, L, I, T, Y}$. Two cards are drawn at random without replacement. Let&nbsp;$A$ be the event that the first card drawn is $A$, and $B$ the event that the second card drawn is $B$.</p>
<ol class="abc-list"><li data-list-item-id="ibsl-q016-part-1-003" data-list-item-path="ibsl-q016-part-1-003"><div class="list-item-paragraph">Find $P(A)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="ibsl-q016-part-2-004" data-list-item-path="ibsl-q016-part-2-004"><div class="list-item-paragraph">Find $P(B|A)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="ibsl-q016-part-3-005" data-list-item-path="ibsl-q016-part-3-005"><div class="list-item-paragraph">Find $P(A ∩ B)$.<span class="inline-marks">[3]</span></div></li></ol>

The letters of the word $\mathrm{PROBABILITY}$ are written on $11$ cards: $\mathrm{P, R, O, B, A, B, I, L, I, T, Y}$. Two cards are drawn at random without replacement. Let $A$ be the event that the first card drawn is $A$, and $B$ the event that the second card drawn is $B$.

Calculate ordered draws without replacement and show dependence.

<p>In any given season, a soccer team plays $65 \%$ of their games at home. When the team plays at home, they win $83 \%$ of their games. When they play away from home, they win $26 \%$ of their games. The team plays one game.</p>
<ol class="abc-list"><li data-list-item-id="ibsl-q019-part-1-003" data-list-item-path="ibsl-q019-part-1-003"><div class="list-item-paragraph">Find the probability that the team wins the game.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="ibsl-q019-part-2-004" data-list-item-path="ibsl-q019-part-2-004"><div class="list-item-paragraph">If the team does not win the game, find the probability that the game was played at home.<span class="inline-marks">[4]</span></div></li></ol>

In any given season, a soccer team plays $65 \%$ of their games at home. When the team plays at home, they win $83 \%$ of their games. When they play away from home, they win $26 \%$ of their games. The team plays one game.

Find the probability that the team wins the game.

If the team does not win the game, find the probability that the game was played at home.

Use total and conditional probability for home/away match outcomes.

<p>At a shopping mall, one ticket is given to every customer who spends $\$1000$ or more. With the ticket, the customer will be invited to draw a ball from the lucky draw box containing $35$ white and $5$ orange balls.</p>
<p>If the first draw is an orange ball, the customer wins a special prize. The draw ends and the ball is not replaced.</p>
<p>If the first draw is a white ball, the ball is replaced in the box and the customer is invited to draw a second time.</p>
<p>If the second draw is an orange ball, the customer wins the special prize. The draw ends and the ball is not replaced.</p>
<p>If the second draw is a white ball, the ball is replaced in the box and customer walks away with nothing.</p>
<p>The lucky draw is over when all the $5$ orange balls have been drawn by customers.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">If none of the special prizes have been won yet, find the probability that the customer with the ticket</div><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-paragraph">draws $2$ white balls,</div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-paragraph">wins the special prize.</div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">If only one orange ball is left, find the probability of winning the special prize.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">By considering your answers and the format of the lucky draw, will it take the next customer a longer/shorter/same amount of time to win the special prize?<br>Briefly explain your answer.</div></li></ol>

At a shopping mall, one ticket is given to every customer who spends $\$1000$ or more. With the ticket, the customer will be invited to draw a ball from the lucky draw box containing $35$ white and $5$ orange balls.

If the first draw is an orange ball, the customer wins a special prize. The draw ends and the ball is not replaced.

If the first draw is a white ball, the ball is replaced in the box and the customer is invited to draw a second time.

If the second draw is an orange ball, the customer wins the special prize. The draw ends and the ball is not replaced.

If the second draw is a white ball, the ball is replaced in the box and customer walks away with nothing.

The lucky draw is over when all the $5$ orange balls have been drawn by customers.

If none of the special prizes have been won yet, find the probability that the customer with the ticket

If only one orange ball is left, find the probability of winning the special prize.

Complementary Events

Waiting-Time and Stopping-Rule Probabilities

Probability in a Two-Draw Lucky Dip

<p>A bag contains $10$ identical balls of which $4$ are red and $6$ are blue. Peter and Glen take turns to draw a ball from the bag without replacement. The first student to take a blue ball wins the game and the game ends.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Given that Peter takes the first draw, Glen takes the second draw, Peter then takes the third draw and so on, state, as a fraction, the probability that</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Peter wins at the first draw.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Glen wins at the second draw.</span></div></li><li><div class="list-item-content"><span class="list-item-text">no fourth draw is required.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">If the game ends at the $n^{\mathrm{th}}$ draw, state the largest value of $n$.</span></div></li></ol>

A bag contains $10$ identical balls of which $4$ are red and $6$ are blue. Peter and Glen take turns to draw a ball from the bag without replacement. The first student to take a blue ball wins the game and the game ends.

Given that Peter takes the first draw, Glen takes the second draw, Peter then takes the third draw and so on, state, as a fraction, the probability that

If the game ends at the $n^{\mathrm{th}}$ draw, state the largest value of $n$.

Sampling Without Replacement with a Stopping Rule

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTg3MDE2NjQtMXR2YnFhMWg0NXcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTYzOCwiZXhwIjoxNzkyNzQ5OTM4fQ.YjWYanwKFHHxKOF6C5vnHr_cgD2JcVBvwkZRHfcRrew" 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>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.

Draw a tree diagram to illustrate the experiment.

Probabilities of Successive Events

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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjYxNDQxNjM2MjgtYWpjcTFuZzI0M2wucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTYzOCwiZXhwIjoxNzkyNzQ5OTM4fQ.3vzsmMdgqUWdBbbvVL_BfhrVpWXBXGr62KAHUnovqYo" 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 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>There are $2$ bags, $A$ and $B$. $A$ contains $3$ red balls and $5$ green balls while $B$ contains $4$ red balls and $3$ green balls. A game is played by $2$ players where a fair $6$ sided die is first thrown.<br></p>
<p>If the die shows the numbers $1$ or $2$, the player picks a ball from Bag $A$ and places it in Bag $B$. He then picks a ball from Bag $B$. If the die shows $3$, $4$, $5$ or $6$, the player loses his turn. The other player will then throw the die. The player who picks a red ball from Bag $B$ wins the game. Mark and Anthony take turns to play the game with Mark starting first. After each person has played the game, the balls are placed back into their original bags and the game starts again.</p>
<p>The game stops when a player has won.</p>
<p>Find the probability that</p>
<ol class="abc-list"><li data-list-item-id="item-1766239416643-9a8qfpxv4" data-list-item-path="item-1766239416643-9a8qfpxv4"><div class="list-item-paragraph">Mark wins the game on the first try.</div></li><li data-list-item-id="item-1766239417625-iuw4tlw34" data-list-item-path="item-1766239417625-iuw4tlw34"><div class="list-item-paragraph">Anthony wins the game on his first try.</div></li></ol>
<p>Every ball selection is uniform at random. Each die throw is independent, and the bags are reset before the next turn if nobody has won.</p>

There are $2$ bags, $A$ and $B$. $A$ contains $3$ red balls and $5$ green balls while $B$ contains $4$ red balls and $3$ green balls. A game is played by $2$ players where a fair $6$ sided die is first thrown.


If the die shows the numbers $1$ or $2$, the player picks a ball from Bag $A$ and places it in Bag $B$. He then picks a ball from Bag $B$. If the die shows $3$, $4$, $5$ or $6$, the player loses his turn. The other player will then throw the die. The player who picks a red ball from Bag $B$ wins the game. Mark and Anthony take turns to play the game with Mark starting first. After each person has played the game, the balls are placed back into their original bags and the game starts again.

Every ball selection is uniform at random. Each die throw is independent, and the bags are reset before the next turn if nobody has won.

2020 ACSI EM S4 P1

<p>In a game, two boxes have the following contents.</p>
<p style="text-align: center">Box $ \; A \; $ contains $ \; 3 \; $ green balls and $ \; 4 \; $ white balls.</p>
<p style="text-align: center">Box $ \; B \; $ contains $ \; 6 \; $ green balls and $ \; 5 \; $ white balls.</p>
<p>A fair die is thrown once. Two balls are then drawn in sequence in the following manner:</p>
<p style="margin-left: 2rem">1. If the number that appears on the top face of the die is less than $3$, the first ball is drawn from Box $A$ and transferred to Box $B$. Otherwise, the first ball is drawn from Box $B$ and transferred to Box $A$.</p>
<p style="margin-left: 2rem">2. The second ball is then randomly picked from the box that received the transferred ball in step $1$. </p>
<ol class="i-list"><li data-list-item-id="item-1759834152889-22rkq0cxa" data-list-item-path="item-1759834152889-22rkq0cxa"><div class="list-item-paragraph">Draw a probability tree diagram to represent the above information.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759834153790-7awtnm2mp" data-list-item-path="item-1759834153790-7awtnm2mp"><div class="list-item-paragraph">If the second ball drawn is white, the player wins the game. Otherwise, the player loses the game. Find the probability that the first ball drawn is from Box $A$, given that the player wins the game.<span class="inline-marks">[5]</span></div></li></ol>

In a game, two boxes have the following contents.

Box $ \; A \; $ contains $ \; 3 \; $ green balls and $ \; 4 \; $ white balls.

Box $ \; B \; $ contains $ \; 6 \; $ green balls and $ \; 5 \; $ white balls.

A fair die is thrown once. Two balls are then drawn in sequence in the following manner:

1. If the number that appears on the top face of the die is less than $3$, the first ball is drawn from Box $A$ and transferred to Box $B$. Otherwise, the first ball is drawn from Box $B$ and transferred to Box $A$.

2. The second ball is then randomly picked from the box that received the transferred ball in step $1$. 

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