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 Venn diagram below shows events A and B where&nbsp;$P(A) = 0.3$, $P(A ∪ B) = 0.6$ and $P(A ∩ B) = 0.1$. The values $m, n, p$ and $q$ are probabilities.</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-q003-part-1-014" data-list-item-path="ibsl-q003-part-1-014"><ol class="i-list"><li data-list-item-id="ibsl-q003-part-1-1-015" data-list-item-path="ibsl-q003-part-1-014/ibsl-q003-part-1-1-015"><div class="list-item-paragraph"><p>Write down the value of $n$.</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="ibsl-q003-part-1-2-016" data-list-item-path="ibsl-q003-part-1-014/ibsl-q003-part-1-2-016"><div class="list-item-paragraph"><p>Find the value of $m$, of $p$, and of $q$.</p></div></li></ol></li><li data-list-item-id="ibsl-q003-part-2-017" data-list-item-path="ibsl-q003-part-2-017"><div class="list-item-paragraph"><p>Find $P(B')$.</p><span class="inline-marks">[2]</span></div></li></ol>

The Venn diagram below shows events A and B where $P(A) = 0.3$, $P(A ∪ B) = 0.6$ and $P(A ∩ B) = 0.1$. The values $m, n, p$ and $q$ are probabilities.

Set Theory and Probability — Combined Problems

Complementary Events

Complete Venn-region probabilities and calculate a complement.

Probability

<p>Consider the events $A$ and $B$, where&nbsp;$P(A)=\frac25$, $P(B')=\frac14$ and $P(A\cup B)=\frac78$.</p>
<ol class="abc-list"><li data-list-item-id="ibsl-q044-part-1-003" data-list-item-path="ibsl-q044-part-1-003"><div class="list-item-paragraph">Write down $P(B)$.</div></li><li data-list-item-id="ibsl-q044-part-2-004" data-list-item-path="ibsl-q044-part-2-004"><div class="list-item-paragraph">Find $P(A ∩ B)$.</div></li><li data-list-item-id="ibsl-q044-part-3-005" data-list-item-path="ibsl-q044-part-3-005"><div class="list-item-paragraph">Find $P(A | B)$.</div></li></ol>
<p><span class="inline-marks">[6]</span></p>

Consider the events $A$ and $B$, where $P(A)=\frac25$, $P(B')=\frac14$ and $P(A\cup B)=\frac78$.

Use complements, unions, intersections and conditional probability for two events.

<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="eyJheGlzIjpmYWxzZSwiZ3JpZCI6ZmFsc2UsIndpZHRoIjo1MjAsImhlaWdodCI6MjYwLCJlbGVtZW50cyI6W3siaWQiOiJpYnNsLXEwMDktdS0wMDIiLCJ0eXBlIjoicG9seWdvbiIsImNvbG9yIjoiIzExMTgyNyIsImZpeGVkIjp0cnVlLCJwb2ludHMiOltbLTUsM10sWzUsM10sWzUsLTNdLFstNSwtM11dLCJmaWxsT3BhY2l0eSI6MH0seyJpZCI6Imlic2wtcTAwOS1hLTAwMyIsInR5cGUiOiJlbGxpcHNlIiwiY29sb3IiOiIjMTExODI3IiwiZml4ZWQiOnRydWUsImNlbnRlciI6Wy0xLjQsMF0sInJhZGl1c1giOjIuMywicmFkaXVzWSI6MS44LCJmaWxsT3BhY2l0eSI6MH0seyJpZCI6Imlic2wtcTAwOS1iLTAwNCIsInR5cGUiOiJlbGxpcHNlIiwiY29sb3IiOiIjMTExODI3IiwiZml4ZWQiOnRydWUsImNlbnRlciI6WzEuNCwwXSwicmFkaXVzWCI6Mi4zLCJyYWRpdXNZIjoxLjgsImZpbGxPcGFjaXR5IjowfSx7IngiOi0zLjEsInkiOjIuMjUsImlkIjoiaWJzbC1xMDA5LWxhYmVsLTAwNSIsInR5cGUiOiJ0ZXh0IiwiY29sb3IiOiIjMTExODI3IiwiZml4ZWQiOnRydWUsImxhYmVsIjoiQSJ9LHsieCI6Mi40NSwieSI6Mi4yNSwiaWQiOiJpYnNsLXEwMDktbGFiZWwtMDA2IiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMxMTE4MjciLCJmaXhlZCI6dHJ1ZSwibGFiZWwiOiJCIn0seyJ4IjotNC42LCJ5IjoyLjYsImlkIjoiaWJzbC1xMDA5LWxhYmVsLTAwNyIsInR5cGUiOiJ0ZXh0IiwiY29sb3IiOiIjMTExODI3IiwiZml4ZWQiOnRydWUsImxhYmVsIjoiVSJ9LHsieCI6LTIuMSwieSI6MCwiaWQiOiJpYnNsLXEwMDktbGFiZWwtMDA4IiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMxMTE4MjciLCJmaXhlZCI6dHJ1ZSwibGFiZWwiOiJwIn0seyJ4IjowLCJ5IjowLCJpZCI6Imlic2wtcTAwOS1sYWJlbC0wMDkiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzExMTgyNyIsImZpeGVkIjp0cnVlLCJsYWJlbCI6IjAuMyJ9LHsieCI6Mi4xLCJ5IjowLCJpZCI6Imlic2wtcTAwOS1sYWJlbC0wMTAiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzExMTgyNyIsImZpeGVkIjp0cnVlLCJsYWJlbCI6InEifSx7IngiOjQuMiwieSI6LTIuMiwiaWQiOiJpYnNsLXEwMDktbGFiZWwtMDExIiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMxMTE4MjciLCJmaXhlZCI6dHJ1ZSwibGFiZWwiOiJyIn1dLCJzaG93WEF4aXMiOmZhbHNlLCJzaG93WUF4aXMiOmZhbHNlLCJ4QXhpc0xhYmVsIjoiIiwieUF4aXNMYWJlbCI6IiIsImJvdW5kaW5nQm94IjpbLTYsNCw2LC00XSwib3JpZ2luTGFiZWwiOiIiLCJzaG93QXhpc0xhYmVscyI6ZmFsc2UsInNob3dOYXZpZ2F0aW9uIjpmYWxzZSwia2VlcEFzcGVjdFJhdGlvIjp0cnVlfQ==" 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 $r$.

Independent 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.

Conditional Probability

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>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

Probability Using Permutations and Combinations

Constructing and Using Probability Trees

Conditional Probability from Tree Diagrams

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.

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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3MDEwMDM3NTEtcWkzbXNzdjY1aC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzQ5Njc0LCJleHAiOjE3OTI3NDk5NzR9.BfcvzW8U9aJpV4dDoUydyorpmbxNVYHw6pl94I6tfrU" 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>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.

Dependent 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTg3MDE2NjQtMXR2YnFhMWg0NXcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTY3NCwiZXhwIjoxNzkyNzQ5OTc0fQ.CohhjuaorSfo7ikH0jcRxQDTVnJVS115N3e-TBUtqS8" 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>Ariel and Ben are taking part in a swimming race. There can only be one winner. The probability of Ariel winning the race is $\frac{1}{3}$. The probability of Ben winning the race is $\frac{1}{5}$. Find the probability that</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">either Ariel or Ben wins,</span></div></li><li><div class="list-item-content"><span class="list-item-text">neither Ariel nor Ben wins.</span></div></li></ol>

Ariel and Ben are taking part in a swimming race. There can only be one winner. The probability of Ariel winning the race is $\frac{1}{3}$. The probability of Ben winning the race is $\frac{1}{5}$. Find the probability that

Mutually Exclusive Events

Mutually Exclusive Events: Axioms of Probability and Summation

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

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

Construct a possibility diagram showing all the possible outcomes.

Probability Using Possibility Diagrams

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

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

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

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

the two coins do not have the same value.

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

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

Probability Analysis: Determining Unknown Frequencies via Algebraic Equations

<p>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.

Probabilities of Successive Events

2020 HCI EM S4 P2

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

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

Probability: Drawing Balls from a Bag (Algebraic Form)

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

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

Probability of a Single Event

Probability: Choosing a Fish from a Tank

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765333907233-pmxyzzhpzai.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUzMzM5MDcyMzMtcG14eXp6aHB6YWkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTY3NCwiZXhwIjoxNzkyNzQ5OTc0fQ.vrrv6BoOjhcr7fqxdBit2TDYQ22_aNKRutiTIVWkxp8" style="max-width: min(200px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The pie chart above shows a circle divided into sectors of different colours.  A point is selected randomly in the circle.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Calculate the probability that the point lies in the Red sector.</div></li><li><div class="list-item-paragraph">Calculate the probability that the point does not lie in the Blue sector.</div></li><li><div class="list-item-paragraph">Given the probability that the point lies in the Yellow sector is $\frac{3}{10}$, find the value of $x$.</div></li><li><div class="list-item-paragraph">Hence, calculate the probability that the point lies in the Black sector.</div></li></ol>

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

Probability: Reading a Sector Diagram

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

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

Math Question Bank