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 shows a circle divided into three sectors A, B and C with central angles $90^\circ$, $120^\circ$ and $150^\circ$, respectively. Sectors A and B are shaded. The arrow is spun and cannot land on a boundary. Let&nbsp;$A$, $B$, $C$ and $S$ denote landing in sector A, B, C and a shaded region, respectively.</p>
<div class="graph-block" data-graph-config="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" data-width="330" data-height="300" style="width:330px;height:300px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="abc-list"><li data-list-item-id="ibsl-q077-part-1-013" data-list-item-path="ibsl-q077-part-1-013"><div class="list-item-paragraph">$P(B)$;</div></li><li data-list-item-id="ibsl-q077-part-2-014" data-list-item-path="ibsl-q077-part-2-014"><div class="list-item-paragraph">$P(S)$;</div></li><li data-list-item-id="ibsl-q077-part-3-015" data-list-item-path="ibsl-q077-part-3-015"><div class="list-item-paragraph">$P(A|S)$.</div></li></ol>
<p><span class="inline-marks">[6]</span></p>

The diagram shows a circle divided into three sectors A, B and C with central angles $90^\circ$, $120^\circ$ and $150^\circ$, respectively. Sectors A and B are shaded. The arrow is spun and cannot land on a boundary. Let $A$, $B$, $C$ and $S$ denote landing in sector A, B, C and a shaded region, respectively.

Set Theory and Probability — Combined Problems

Conditional Probability

Probability of a Single Event

Calculate spinner-sector, shaded-region and conditional probabilities.

Probability

<p>One hundred students were asked how many hand phones they have in their home. The results of the survey are summarized in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">No. of hand phones ($x$)</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">No. of students ($f$)</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$fx$</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$1$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$15$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$c$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$2$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$25$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$50$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$3$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$b$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$d$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$4$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$a$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$84$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$12$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$60$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$6$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$30$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Total</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$100$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$e$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the values of $a$, $b$, $c$, $d$ and $e$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">the mean and standard deviation of the number of hand phones in their home.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">There are $5$ red cards, $6$ yellow cards and $4$ blue cards in a box. Annie draws a card from the box at random. Write down the probability that she will not get a blue card.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Brandon draws $2$ cards at random, one after the other, without replacement. Find the probability that the two cards are of different colour.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Annie and Brandon take turn to draw one card at a time, without replacement, until either player gets a blue card. Assuming that Annie starts first, find the probability that she will get a blue card exactly on her second try.</span></div></li></ol>

One hundred students were asked how many hand phones they have in their home. The results of the survey are summarized in the table below.

There are $5$ red cards, $6$ yellow cards and $4$ blue cards in a box. Annie draws a card from the box at random. Write down the probability that she will not get a blue card.

Brandon draws $2$ cards at random, one after the other, without replacement. Find the probability that the two cards are of different colour.

Annie and Brandon take turn to draw one card at a time, without replacement, until either player gets a blue card. Assuming that Annie starts first, find the probability that she will get a blue card exactly on her second try.

Mean and Standard Deviation for Ungrouped Data

Probabilities of Successive Events

Constructing and Interpreting Frequency Tables

Waiting-Time and Stopping-Rule Probabilities

Statistics

<p>A teacher recorded the individual time taken by $200$ students to complete a cross-country race. The results are presented in the form of a cumulative frequency curve.</p>
<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/1769330452944-u09rnsjaqz.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzA0NTI5NDQtdTA5cm5zamFxei5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzQ5NzMyLCJleHAiOjE3OTI3NTAwMzJ9.5HGAeIk6RdRpFQnp18PbZxWpEsMAVmNwx3vz6jblGzY" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use the curve to estimate</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the median time taken to run the race,</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the interquartile range of the distribution,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the percentage of students who took longer than $66$ min to finish the race,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the probability of choosing a student who took $45$ min or less time to finish the race.</span><span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">The frequency table relates to the same cross-country race results.</span></div></li></ol>
<div class="content-block content-block--table"><div class="table-container"><table><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Frequency</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Time ($t$ min)</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$13$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$30 &lt; t \le 40$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$r$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$40 &lt; t \le 50$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$90$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$50 &lt; t \le 60$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$s$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$60 &lt; t \le 70$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$9$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$70 &lt; t \le 80$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$4$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$80 &lt; t \le 90$</td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the value of $r$ and the value of $s$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Showing your method clearly, estimate the mean time required to run the race.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Estimate the standard deviation of the distribution.</span><span class="inline-marks">[2]</span></div></li></ol>

A teacher recorded the individual time taken by $200$ students to complete a cross-country race. The results are presented in the form of a cumulative frequency curve.

The frequency table relates to the same cross-country race results.

Find the value of $r$ and the value of $s$.

Showing your method clearly, estimate the mean time required to run the race.

Estimate the standard deviation of the distribution.

Percentiles and Quartiles from Cumulative Frequency Curves

Cumulative Frequency Table and Curve

Mean and Standard Deviation for Grouped Data

<p>A dart board made up of $3$ circles with $3$ different colours is shown in the diagram. Each circle has a radius of $2$ cm, $6$ cm and $8$ cm respectively. Assume that all darts thrown hit the board.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li data-list-item-id="item-1773731735577-c4upj30qs" data-list-item-path="item-1773731735577-c4upj30qs"><div class="list-item-paragraph">Show that the probability of a dart hitting the red area is $\frac{1}{16}$.</div></li><li data-list-item-id="item-1773731736094-cpv90njoh" data-list-item-path="item-1773731736094-cpv90njoh"><div class="list-item-paragraph">If&nbsp;$2$ darts are thrown, one at a time, find the probability of both darts hitting the orange area.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767699347486-1b2bgfgqzq3.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTkzNDc0ODYtMWIyYmdmZ3F6cTMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTczMiwiZXhwIjoxNzkyNzUwMDMyfQ.Y7m3WbkIbeQX9r7Pnlm8LLoqUGp9MooZIVikEyIzhSA" style="max-width: min(240px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>
<p>Assume each dart lands uniformly over the area of the board, and the two throws are independent.</p>

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

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

Independent Events

Circumference and Area of Circles

Geometric Probability on a Dartboard

Perimeter and Area of Plane Figures

<p>A two-digit number is formed at random using the digits $2$, $3$, $4$ and $5$ with repetition of digits allowed. Find the probability of forming</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">an odd number,</span></div></li><li><div class="list-item-content"><span class="list-item-text">forming a number $x$, where $x &gt; 44$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">forming a number divisible by $5$.</span></div></li></ol>

A two-digit number is formed at random using the digits $2$, $3$, $4$ and $5$ with repetition of digits allowed. Find the probability of forming

forming a number $x$, where $x > 44$,

Calculating Probability in Uniform Sample Spaces

<p>A fair $6$-sided dice is thrown. Find the probability that the number shown is</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">an even number greater than $2$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">a prime number,</span></div></li><li><div class="list-item-content"><span class="list-item-text">a factor of $6$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">an $8$.</span></div></li></ol>

A fair $6$-sided dice is thrown. Find the probability that the number shown is

Probability of a Single Event: Discrete Outcomes

<p>The following data shows the number of hours $24$ students of a class spent studying per day.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 400px;"><colgroup><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p>$4$</p></th><th style="text-align: center; --cell-justify: center"><p>$3$</p></th><th style="text-align: center; --cell-justify: center"><p>$4$</p></th><th style="text-align: center; --cell-justify: center"><p>$6$</p></th><th style="text-align: center; --cell-justify: center"><p>$1$</p></th><th style="text-align: center; --cell-justify: center"><p>$4$</p></th><th style="text-align: center; --cell-justify: center"><p>$0$</p></th><th style="text-align: center; --cell-justify: center"><p>$5$</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>$3$</p></td><td style="text-align: center; --cell-justify: center"><p>$1$</p></td><td style="text-align: center; --cell-justify: center"><p>$4$</p></td><td style="text-align: center; --cell-justify: center"><p>$4$</p></td><td style="text-align: center; --cell-justify: center"><p>$1$</p></td><td style="text-align: center; --cell-justify: center"><p>$4$</p></td><td style="text-align: center; --cell-justify: center"><p>$5$</p></td><td style="text-align: center; --cell-justify: center"><p>$6$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$0$</p></td><td style="text-align: center; --cell-justify: center"><p>$3$</p></td><td style="text-align: center; --cell-justify: center"><p>$4$</p></td><td style="text-align: center; --cell-justify: center"><p>$0$</p></td><td style="text-align: center; --cell-justify: center"><p>$4$</p></td><td style="text-align: center; --cell-justify: center"><p>$0$</p></td><td style="text-align: center; --cell-justify: center"><p>$3$</p></td><td style="text-align: center; --cell-justify: center"><p>$3$</p></td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Using the grid below, draw a dot diagram to represent the above data.	</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765431475759-62j12cvscbf.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjU0MzE0NzU3NTktNjJqMTJjdnNjYmYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTczMiwiZXhwIjoxNzkyNzUwMDMyfQ.b_rb4lQQK7M30VE5Yrp_JADs72Nz6fxLOzlT3IVCR7E" alt="List item image" class="aligncenter custom-width" style="max-width: min(350px, 100%); height: auto;" /></div></li><li><div class="list-item-paragraph">What is the modal number of hours the students spent studying?</div></li><li><div class="list-item-paragraph">Calculate the mean number of hours the students spent studying.</div></li><li><div class="list-item-paragraph">If the above data was presented on a pie chart, what is the size of the angle that represents the number of students who study at least $3$ hours a day?</div></li><li><div class="list-item-paragraph">What is the probability of a student studying $4$ hours a day?	</div></li></ol>

The following data shows the number of hours $24$ students of a class spent studying per day.

Mean of a Frequency Distribution

Pie Charts

Mode

Dot Diagrams

Number of Hours Spent Studying: Mode, Mean, and Probability

<p>The table below shows the number of cars per household.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 360px;"><colgroup><col style="width: 110px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p><strong>Number of cars</strong></p></th><th style="text-align: center; --cell-justify: center"><p>$0$</p></th><th style="text-align: center; --cell-justify: center"><p>$1$</p></th><th style="text-align: center; --cell-justify: center"><p>$2$</p></th><th style="text-align: center; --cell-justify: center"><p>$3$</p></th><th style="text-align: center; --cell-justify: center"><p>$4$</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p><strong>Number of households</strong></p></td><td style="text-align: center; --cell-justify: center"><p>$6$</p></td><td style="text-align: center; --cell-justify: center"><p>$5$</p></td><td style="text-align: center; --cell-justify: center"><p>$3$</p></td><td style="text-align: center; --cell-justify: center"><p>$x$</p></td><td style="text-align: center; --cell-justify: center"><p>$1$</p></td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-paragraph">If the median number of cars is $1$, find the possible values of $x$.</div></li><li><div class="list-item-paragraph">If&nbsp;$x=5$,</div><ol class="i-list"><li><div class="list-item-paragraph">state the modal number of cars,</div></li><li><div class="list-item-paragraph">find the probability of choosing a household with at least $2$ cars.</div></li></ol></li></ol>

The table below shows the number of cars per household.

Finding Unknowns from Averages and Spread

Household Car Ownership: Median, Mode, and Probability

<p>The following data shows the number of customers across $12$ restaurants at a given moment.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 400px;"><colgroup><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p>$15$</p></th><th style="text-align: center; --cell-justify: center"><p>$15$</p></th><th style="text-align: center; --cell-justify: center"><p>$18$</p></th><th style="text-align: center; --cell-justify: center"><p>$16$</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>$12$</p></td><td style="text-align: center; --cell-justify: center"><p>$16$</p></td><td style="text-align: center; --cell-justify: center"><p>$16$</p></td><td style="text-align: center; --cell-justify: center"><p>$15$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$13$</p></td><td style="text-align: center; --cell-justify: center"><p>$14$</p></td><td style="text-align: center; --cell-justify: center"><p>$16$</p></td><td style="text-align: center; --cell-justify: center"><p>$12$</p></td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Using the grid below, draw a dot diagram to represent the above data.	</div></li><li><div class="list-item-paragraph">Calculate the mean number of customers across the restaurants.</div></li><li><div class="list-item-paragraph">If the above data was presented on a pie chart, what is the size of the angle that represents the number of restaurants that have at most $14$ customers?</div></li><li><div class="list-item-paragraph">What is the probability of a restaurant having $15$ customers at a given moment?</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765356605664-r1hvseay03d.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUzNTY2MDU2NjQtcjFodnNlYXkwM2QucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTczMiwiZXhwIjoxNzkyNzUwMDMyfQ.t05E_mQNCm07fI6PCwJe8P6i3tkL48nzQAIBoxKtoBw" alt="List item image" class="aligncenter custom-width" style="max-width: min(400px, 100%); height: auto;" /></div></li></ol>

The following data shows the number of customers across $12$ restaurants at a given moment.

Number of Customers Across Restaurants

<p>The following data shows the number of hours $24$ students of a class spent studying per day.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 560px;"><colgroup><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p>$3$</p></th><th style="text-align: center; --cell-justify: center"><p>$3$</p></th><th style="text-align: center; --cell-justify: center"><p>$4$</p></th><th style="text-align: center; --cell-justify: center"><p>$6$</p></th><th style="text-align: center; --cell-justify: center"><p>$1$</p></th><th style="text-align: center; --cell-justify: center"><p>$4$</p></th><th style="text-align: center; --cell-justify: center"><p>$0$</p></th><th style="text-align: center; --cell-justify: center"><p>$2$</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>$3$</p></td><td style="text-align: center; --cell-justify: center"><p>$1$</p></td><td style="text-align: center; --cell-justify: center"><p>$0$</p></td><td style="text-align: center; --cell-justify: center"><p>$4$</p></td><td style="text-align: center; --cell-justify: center"><p>$5$</p></td><td style="text-align: center; --cell-justify: center"><p>$4$</p></td><td style="text-align: center; --cell-justify: center"><p>$5$</p></td><td style="text-align: center; --cell-justify: center"><p>$6$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$0$</p></td><td style="text-align: center; --cell-justify: center"><p>$3$</p></td><td style="text-align: center; --cell-justify: center"><p>$2$</p></td><td style="text-align: center; --cell-justify: center"><p>$0$</p></td><td style="text-align: center; --cell-justify: center"><p>$3$</p></td><td style="text-align: center; --cell-justify: center"><p>$0$</p></td><td style="text-align: center; --cell-justify: center"><p>$3$</p></td><td style="text-align: center; --cell-justify: center"><p>$3$</p></td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Using the grid below, draw a dot diagram to represent the above data.	</div></li><li><div class="list-item-paragraph">Calculate the mean number of hours the students spent studying.</div></li><li><div class="list-item-paragraph">If the above data was presented on a pie chart, what is the size of the angle that represents the number of students who study at least $3$ hours a day?</div></li><li><div class="list-item-paragraph">What is the probability of a student studying $4$ hours a day?	</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765353365573-827m1wboxzo.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUzNTMzNjU1NzMtODI3bTF3Ym94em8ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTczMiwiZXhwIjoxNzkyNzUwMDMyfQ.NqYDSIBUPeax-tEeqB2GzhG0rN14O1Pt8o1D6cVJ2Kw" alt="List item image" class="aligncenter custom-width" style="max-width: min(400px, 100%); height: auto;" /></div></li></ol>

Finding The Mean From A Dot Diagram

<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.

Complementary Events

Finding Unknowns in Probability Models

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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUzMzM5MDcyMzMtcG14eXp6aHB6YWkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTczMiwiZXhwIjoxNzkyNzUwMDMyfQ.yNmO06tBL8mR_rta2t2sxsuR0srLGvQ4-0WKd-Mxs90" 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>The digits $2,6,7$ are on three separate cards. Choose uniformly from all the distinct one- or two-digit numbers that can be formed without reusing a card. Find the probability that the chosen number</p>
<ol class="abc-list"><li data-list-item-id="item-1765279275639-99u1zr8e9" data-list-item-path="item-1765279275639-99u1zr8e9"><div class="list-item-paragraph">consists of two digits,</div></li><li data-list-item-id="item-1765279276989-hpnagfnij" data-list-item-path="item-1765279276989-hpnagfnij"><div class="list-item-paragraph">is a prime number,</div></li><li data-list-item-id="item-1765279277583-ld3rhibv1" data-list-item-path="item-1765279277583-ld3rhibv1"><div class="list-item-paragraph">is a multiple of $5$.</div></li></ol>

The digits $2,6,7$ are on three separate cards. Choose uniformly from all the distinct one- or two-digit numbers that can be formed without reusing a card. Find the probability that the chosen number

Probability: Forming a One- or Two-Digit Number

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

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

Probability: Picking a Ball from a Bag

<p>In a class of 40 students, ${\frac{2}{5}}$ are girls and the remaining students are boys. Two of the girls are short-sighted compared to 5 boys who are short-sighted. If a student is selected at random, find the probability that the student will be</p>
<ol class="abc-list"><li><div class="list-item-paragraph">a short-sighted boy,</div></li><li><div class="list-item-paragraph">a non short-sighted girl.</div></li></ol>

In a class of 40 students, ${\frac{2}{5}}$ are girls and the remaining students are boys. Two of the girls are short-sighted compared to 5 boys who are short-sighted. If a student is selected at random, find the probability that the student will be

Probability: Selecting a Student from a Class

<p>There are $47$ cards in a deck of which $22$ are picture cards and $25$ are number cards. Each card has a suit of either a clubs or a heart. There is a total of $27$ cards with the suit of clubs, $18$ of which are number cards. </p>
<p>A card is chosen at random from the deck, find the probability of choosing</p>
<ol class="i-list"><li><div class="list-item-paragraph">a picture card,</div></li><li><div class="list-item-paragraph">a number card with a suit of clubs,</div></li><li><div class="list-item-paragraph">a picture card with a suit of heart,</div></li><li><div class="list-item-paragraph">a picture card or a card with a suit of heart.</div></li></ol>

There are $47$ cards in a deck of which $22$ are picture cards and $25$ are number cards. Each card has a suit of either a clubs or a heart. There is a total of $27$ cards with the suit of clubs, $18$ of which are number cards. 

A card is chosen at random from the deck, find the probability of choosing

Addition Rule for Overlapping Events

Basic Probability: Picture and Number Cards

<p>A bag contains $24$ candies of which half of them are orange flavour and the remaining are apple flavour. The candies are wrapped in either black or white paper. There are $8$ candies wrapped in white paper, $5$ of which are apple flavour. </p>
<p>A candy is chosen at random from the bag, find the probability of choosing </p>
<ol class="i-list"><li><div class="list-item-paragraph">a candy with apple flavour,</div></li><li><div class="list-item-paragraph">a candy with orange flavour wrapped in black paper,</div></li><li><div class="list-item-paragraph">a candy with orange flavour or a candy wrapped in black paper. </div></li></ol>

A bag contains $24$ candies of which half of them are orange flavour and the remaining are apple flavour. The candies are wrapped in either black or white paper. There are $8$ candies wrapped in white paper, $5$ of which are apple flavour. 

A candy is chosen at random from the bag, find the probability of choosing 

Probability: Choosing a Candy from a Mixed Collection

<p>An $8$-sided fair die with faces labelled $4, 4, 5, 6, 7, 8, 9, 10$ is rolled once. Find the probability of getting</p>
<ol class="i-list"><li><div class="list-item-paragraph">an even number,</div></li><li><div class="list-item-paragraph">a prime number,</div></li><li><div class="list-item-paragraph">a ‘$4$’ or a ‘$7$’,</div></li><li><div class="list-item-paragraph">a number more than $6$.</div></li></ol>

An $8$-sided fair die with faces labelled $4, 4, 5, 6, 7, 8, 9, 10$ is rolled once. Find the probability of getting

Probability: Rolling an 8-Sided Die

<p>A $6$-sided fair die is rolled once. Find the probability of getting</p>
<ol class="i-list"><li><div class="list-item-paragraph">an odd number,</div></li><li><div class="list-item-paragraph">a composite number,</div></li><li><div class="list-item-paragraph">a number that is divisible by $3$. </div></li></ol>

A $6$-sided fair die is rolled once. Find the probability of getting

Probability: Rolling a Fair Die

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

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

Math Question Bank