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>Kai and Lina play a sequence of rounds. Kai wins the first round with probability $\dfrac{3}{5}$. For every round after the first, the conditional probability that the winner of the preceding round also wins the next round is $\dfrac{2}{3}$. The match ends as soon as a player wins two consecutive rounds.</p>
<ol class="i-list"><li data-list-item-id="gp-q2-i" data-list-item-path="gp-q2-i"><div class="list-item-paragraph">The first four possible winning sequences for Kai are $KK$, $LKK$, $KLKK$ and $LKLKK$, where&nbsp;$K$ denotes a round won by Kai and $L$ denotes a round won by Lina. Find the probability of each sequence.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="gp-q2-ii" data-list-item-path="gp-q2-ii"><div class="list-item-paragraph">Hence, using the sum to infinity of a geometric progression, find the probability that Kai wins the match.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="gp-q2-iii" data-list-item-path="gp-q2-iii"><div class="list-item-paragraph">Given that Lina wins the first round, find the probability that Kai wins the match.<span class="inline-marks">[2]</span></div></li></ol>

Kai and Lina play a sequence of rounds. Kai wins the first round with probability $\dfrac{3}{5}$. For every round after the first, the conditional probability that the winner of the preceding round also wins the next round is $\dfrac{2}{3}$. The match ends as soon as a player wins two consecutive rounds.

Conditional Probability

Repeated Trials with Infinitely Many Possible Outcomes

Probabilities of Successive Events

Application of Sum to Infinity

Waiting-Time and Stopping-Rule Probabilities

Sequence and Series

Probability

<p>Ava and Ben take turns throwing a dart at a target. The probability that Ava hits the target on any throw is $\dfrac{2}{5}$, while the probability that Ben hits it on any throw is $\dfrac{1}{3}$. All throws are independent. A player wins as soon as that player hits the target. Ava throws first.</p>
<ol class="i-list"><li data-list-item-id="gp-q1-i" data-list-item-path="gp-q1-i"><div class="list-item-paragraph">Show that the probability that both players miss in one pair of consecutive throws, one by Ava followed by one by Ben, is $\dfrac{2}{5}$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="gp-q1-ii" data-list-item-path="gp-q1-ii"><div class="list-item-paragraph">Find the probability that Ava wins the game.<span class="inline-marks">[3]</span></div></li></ol>
<p>In a second game, a biased coin is used to choose who throws first. Ben is twice as likely as Ava to throw first. The same hit probabilities and rules apply.</p>
<ol class="i-list" start="3"><li data-list-item-id="gp-q1-iii" data-list-item-path="gp-q1-iii"><div class="list-item-paragraph">Find the probability that Ava wins the second game.<span class="inline-marks">[3]</span></div></li></ol>

Ava and Ben take turns throwing a dart at a target. The probability that Ava hits the target on any throw is $\dfrac{2}{5}$, while the probability that Ben hits it on any throw is $\dfrac{1}{3}$. All throws are independent. A player wins as soon as that player hits the target. Ava throws first.

In a second game, a biased coin is used to choose who throws first. Ben is twice as likely as Ava to throw first. The same hit probabilities and rules apply.

Law of Total 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

Constructing and Interpreting Frequency Tables

Probability of a Single Event

Statistics

<p>The weights of $200$ secondary two pupils at Ace Secondary School are represented by the cumulative frequency curve shown below.</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/1769330982039-i6xy3zqu22q.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzA5ODIwMzktaTZ4eTN6cXUyMnEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTY4NSwiZXhwIjoxNzkyNzQ5OTg1fQ.udC68g9_R5OXm2C-EXT-QUOiUZit7NFbAbkO4SofLbk" style="max-width: min(400px, 100%); height: auto; display: block;" alt="Cumulative Frequency Curve for Weights of Pupils" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use the graph to find</span></div><ol class="i-list"><li><div class="list-item-paragraph">the median weight of the pupils,</div></li><li><div class="list-item-paragraph">the interquartile range,</div></li><li><div class="list-item-paragraph">the probability that at least one of two pupils chosen at random weighs at least $55$ kg.</div></li></ol></li><li><div class="list-item-paragraph">The weights of the pupils are represented in the box-and-whiskers plot shown below. Write down the values of $a$, $b$, $c$, $d$ and $e$.</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/1769331007631-b9z4edd5xs5.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzEwMDc2MzEtYjl6NGVkZDV4czUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTY4NSwiZXhwIjoxNzkyNzQ5OTg1fQ.DgyZUfKaufF0vhJKhPXP5ncKeo_gye_h7aPJO-YlK7Y" alt="List item image" class="aligncenter" style="max-width: 100%; height: auto;" /></div></li><li><div class="list-item-paragraph">The weights of another $200$ secondary two pupils at Best Secondary School are represented in another box-and-whiskers plot shown below. Compare briefly the weights of the secondary two pupils from the two schools.</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/1769331018689-izkd9lsqqf.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzEwMTg2ODktaXprZDlsc3FxZi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzQ5Njg1LCJleHAiOjE3OTI3NDk5ODV9.XwjHcCgzxtD76ppEqgjVOp_9dZMspWbvkZYfsDySOpY" alt="List item image" class="aligncenter" style="max-width: 100%; height: auto;" /></div></li></ol>

The weights of $200$ secondary two pupils at Ace Secondary School are represented by the cumulative frequency curve shown below.

The weights of the pupils are represented in the box-and-whiskers plot shown below. Write down the values of $a$, $b$, $c$, $d$ and $e$.

Percentiles and Quartiles from Cumulative Frequency Curves

Comparing Distributions Using Centre and Spread

Box-and-Whisker Plots

<p>The cumulative frequency curve below illustrates the marks obtained by $120$ students in a Science test for Class $A$.</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/1769330518184-k7awj2ogt0h.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzA1MTgxODQtazdhd2oyb2d0MGgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTY4NSwiZXhwIjoxNzkyNzQ5OTg1fQ.adUPxfCA4gIRMImkB3WfCO3cBvKBH7cT84FYb2LaRiU" style="max-width: min(400px, 100%); height: auto; display: block;" alt="Cumulative frequency curve for Science test marks" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use the graph to find,</span></div><ol class="i-list"><li><div class="list-item-paragraph">the $80^{th}$ percentile.</div></li><li><div class="list-item-paragraph">the interquartile range.</div><div class="list-item-paragraph">In this test, only $15$% of the students scored a distinction.</div></li><li><div class="list-item-paragraph">Find the least marks for a student to be considered for distinction.</div><div class="list-item-paragraph">Two student are chosen at random.</div></li><li><div class="list-item-paragraph">Find the probability that both the students scored above $80$ marks.</div></li></ol></li><li><div class="list-item-paragraph">The cumulative frequency curve can be represented using the following box-and-whisker diagram.</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/1769330604036-b08esn5p68u.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzA2MDQwMzYtYjA4ZXNuNXA2OHUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTY4NSwiZXhwIjoxNzkyNzQ5OTg1fQ.zn_rbrlSfukxXpsoTsjz9zTtYLzMFjEhe1FI-GLPfXs" alt="List item image" class="aligncenter" style="max-width: 100%; height: auto;" /></div><ol class="i-list"><li><div class="list-item-paragraph">State the value of $x_{3}$.</div></li><li><div class="list-item-paragraph">State the value of $x_{5}-x_{1}$. What does $x_{5}-x_{1}$ represent?</div></li></ol></li><li><div class="list-item-paragraph">The box-and-whisker diagram below illustrates the marks obtained by $120$ students in a Science test for Class $B$.</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/1769499239663-n5u3oytoth.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk0OTkyMzk2NjMtbjV1M295dG90aC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzQ5Njg1LCJleHAiOjE3OTI3NDk5ODV9.GpMGETQo8sw-Kq2LvLO7MKlT6k-w3aNWIvp-56c6cDU" alt="List item image" class="aligncenter custom-width" style="max-width: min(400px, 100%); height: auto;" /></div><div class="list-item-paragraph">Compare the test results for the two classes in two different ways.</div></li><li><div class="list-item-paragraph">The results of the science test of Class $C$ of $120$ students have the same median but a smaller standard deviation. Describe how the cumulative frequency curve will differ from the given curve.</div></li></ol>

The cumulative frequency curve below illustrates the marks obtained by $120$ students in a Science test for Class $A$.

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

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

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

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

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

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

Constructing and Using Probability Trees

Conditional Probability from Tree Diagrams

2013 JJC P2 Q9

<p>A bag contains $15$ tokens that are indistinguishable apart from their colours. $2$ of the tokens are blue and the rest are either red or green. Participants are required to draw the tokens randomly, one at a time, from the bag without replacement.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that the probability that a participant draws $2$ red tokens on the first $2$ draws is $\frac{1}{35}$, show that there are $3$ red tokens in the bag.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the probability that a participant draws a red or green token on the second draw.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Events $A$ and $B$ are defined as follows:</div><div class="list-item-paragraph">$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ $A$: A participant draws his/her second red token on the third draw.<br>$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ $B$: A participant draws a blue token on the second draw.</div><div class="list-item-paragraph">Find $\mathrm{P}(A \cup B)$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Determine if&nbsp;$A$ and $B$ are independent events.<span class="inline-marks">[1]</span></div></li></ol>

A bag contains $15$ tokens that are indistinguishable apart from their colours. $2$ of the tokens are blue and the rest are either red or green. Participants are required to draw the tokens randomly, one at a time, from the bag without replacement.

Given that the probability that a participant draws $2$ red tokens on the first $2$ draws is $\frac{1}{35}$, show that there are $3$ red tokens in the bag.

Find the probability that a participant draws a red or green token on the second draw.

Independent Events

Finding Unknowns in Probability Models

Probability Using Permutations and Combinations

Addition Rule for Overlapping Events

2013 HCI P2 Q10

<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/1767944223552-75yqyislj5x.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDQyMjM1NTItNzV5cXlpc2xqNXgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTY4NSwiZXhwIjoxNzkyNzQ5OTg1fQ.VZWkYQlS4cCQTNcOEW2U0mYD37VeaL2uVYsZfrxcayY" style="max-width: min(500px, 100%); height: auto; display: block;" alt="Cumulative frequency graph for Mathematics and Science examinations" class="aligncenter custom-width" /></figure></div>
<p>The cumulative frequency graph shows the distribution of marks for&nbsp;$400$ students in a Mathematics and Science examination.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the median mark for the Mathematics examination.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the interquartile range for the Science examination.</span></div></li><li><div class="list-item-content"><span class="list-item-text">If the distinctions for Mathematics were allocated to the top $25\%$ of the students, find the mark to achieve a distinction.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">One student is chosen at random. You may assume that the marks gained in the two examinations are independent. Expressing your answer as a fraction in its lowest terms, calculate the probability that the student gains</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">less than $20$ marks for both examinations,</span></div></li><li><div class="list-item-content"><span class="list-item-text">more than $60$ marks on one examination, but not on the other.</span></div></li></ol></li></ol>

The cumulative frequency graph shows the distribution of marks for $400$ students in a Mathematics and Science examination.

One student is chosen at random. You may assume that the marks gained in the two examinations are independent. Expressing your answer as a fraction in its lowest terms, calculate the probability that the student gains

Problems Involving Two Cumulative Frequency Curve

Regional Academic Standards: A Comparative Statistical Study

<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.

Dependent 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjYxNDQxNjM2MjgtYWpjcTFuZzI0M2wucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTY4NSwiZXhwIjoxNzkyNzQ5OTg1fQ.YvN0xZmkfHHAgGz0AVO6NfK5fc4TJt6lj4vI3yb8QPY" 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.

Probability Trees for Dependent Events

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

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

Probability of Independent Events: Modeling Multi-Stage Experiments

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

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

Probability of Independent Events: Simultaneous Dice Trials

<p>A 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.

Complementary Events

2020 HCI EM S4 P2

<p>Ava and Barry play a game containing one green marble and two red marbles. Each player intron randomly selects a marble from the bag, notes its colour and replaces it. Ava wins the game if she selects a green marble. Barry wins the game if he selects a red marble. Ava starts the game.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the probability that Ava wins on her first turn.</div></li><li><div class="list-item-paragraph">Find the probability that Barry wins on his first turn.</div></li><li><div class="list-item-paragraph">Find the probability that Ava wins in one of her first three turns.</div></li><li><div class="list-item-paragraph">Find the probability that Ava eventually wins.</div></li></ol>

Ava and Barry play a game containing one green marble and two red marbles. Each player intron randomly selects a marble from the bag, notes its colour and replaces it. Ava wins the game if she selects a green marble. Barry wins the game if he selects a red marble. Ava starts the game.

Worked Example 5

<ol class="abc-list"><li><div class="list-item-paragraph">${A}$, ${B}$ and ${C}$ are taking part in a game in which a prize is hidden in one of $n$ identical closed boxes, where&nbsp;$n &gt; 4$. First, ${A}$ opens at random two boxes in an attempt to find the prize. If&nbsp;${A}$ fails to find the prize, ${B}$ then opens at random two of the remaining boxes in an attempt to find the prize. If&nbsp;${B}$ fails to find the prize, ${C}$ opens all the remaining boxes and finds the prize.</div><ol class="i-list"><li><div class="list-item-paragraph">Find, in terms of $n$, the probability that&nbsp;${A}$ finds the prize.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Show that the probability that B finds the prize is $\frac{2}{n}$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Find the range of values of $n$ for which the probability that&nbsp;${C}$ finds the prize is more than $8$ times the probability that either ${A}$ or ${B}$ find the prize.<span class="inline-marks">[3]</span></div></li></ol></li><li><div class="list-item-paragraph">The events $Q$ and $R$ are such that&nbsp;${P}\left( Q\cap R \right)=x$, $\mathrm{P}\left( Q\cap {R}' \right)=0.1$, ${P}\left( {Q}'\cap R \right)=\mathrm{P}\left( Q \right)$ and ${P}\left( \left( Q\cup R \right)' \right)=\mathrm{P}\left( Q \right)+2\mathrm{P}\left( R \right)$. By using a Venn diagram, or otherwise, find the exact value of $x$.<span class="inline-marks">[3]</span></div></li></ol>

Probability Bounds and Inequalities

N2024 P2 Q7

<p>A store owner receives a shipment of stationery items, including notebooks, pens, and correction tapes. Historical data indicates that&nbsp;$1\%$ of the notebooks, $2\%$ of the pens, and $4\%$ of the correction tapes are defective. The quality of notebooks, pens and correction tapes is independent of one another. The store owner decides to sell the stationery in $180$ packets, each containing one notebook, two pens and one correction tape. A packet is deemed unsatisfactory if any of the four items is defective.</p>
<ol class="i-list"><li data-list-item-id="item-1761026739916-r8mf46f1p" data-list-item-path="item-1761026739916-r8mf46f1p"><div class="list-item-paragraph">Show that the probability that a randomly selected packet is unsatisfactory is $0.0872$, correct to $3$ significant figures.<span class="inline-marks">[2]</span></div></li></ol>
<p>The number of packets that are unsatisfactory is denoted by $X$. You may assume that&nbsp;$X$ can be modelled by a binomial distribution</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1761026778901-9zeslxr7m" data-list-item-path="item-1761026778901-9zeslxr7m"><div class="list-item-paragraph">Find the probability that there are at least $5$ but less than $10$ packets that are unsatisfactory.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761026779456-k1lne558d" data-list-item-path="item-1761026779456-k1lne558d"><div class="list-item-paragraph">Find the least value of $r$ such that the probability that there are more than $r$ packets that are unsatisfactory is at most $0.12$.<span class="inline-marks">[2]</span></div></li></ol>
<p>Before selling the packets of stationery, he decided to select a sample of $9$ packets to check for unsatisfactory packets.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1761026803727-iw5hgizpy" data-list-item-path="item-1761026803727-iw5hgizpy"><div class="list-item-paragraph">How should the packets be selected? Give a reason for this method of selection.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761026804701-qwibu9mih" data-list-item-path="item-1761026804701-qwibu9mih"><div class="list-item-paragraph">Find the probability that the ninth packet is the third unsatisfactory packet selected.<span class="inline-marks">[2]</span></div></li></ol>

A store owner receives a shipment of stationery items, including notebooks, pens, and correction tapes. Historical data indicates that $1\%$ of the notebooks, $2\%$ of the pens, and $4\%$ of the correction tapes are defective. The quality of notebooks, pens and correction tapes is independent of one another. The store owner decides to sell the stationery in $180$ packets, each containing one notebook, two pens and one correction tape. A packet is deemed unsatisfactory if any of the four items is defective.

The number of packets that are unsatisfactory is denoted by $X$. You may assume that $X$ can be modelled by a binomial distribution

Before selling the packets of stationery, he decided to select a sample of $9$ packets to check for unsatisfactory packets.

Binomial Probabilities Using Technology

Binomial Probabilities Using the Probability Mass Function

Population and Random Samples

2024 JPJC P2 Q8

Discrete Random Variables (DRV)

Sampling

<p>National Fruit Company owns a large tomato farm. The tomatoes produced are harvested and sold in boxes of $25$. It is known that $100p\%$ of the tomatoes are rotten. For these boxes, the mean number of rotten tomatoes in a box is $1$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Explain why the context above may not be well-modelled by a binomial distribution.</div><p style="text-align: right;">[1]</p></li></ol>
<p>Assume now that the context above is well-modelled by a binomial distribution.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">State the value of $p$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the probability that a box chosen at random has less than $2$ rotten tomato.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">A customer chose a box and inspected the contents individually. Find the probability that the twenty-first tomato is the fourth rotten tomato and no rotten tomatoes are found subsequently.</div><p style="text-align: right;">[3]</p></li></ol>
<p>Boxes that contains at least $24$ tomatoes that are not rotten are deemed satisfactory.</p>
<ol class="i-list" start="5"><li><div class="list-item-paragraph">A customer first picks $3$ boxes of tomatoes, of which at least $2$ boxes are satisfactory. The customer then decides to buy another $5$ boxes. Find the probability that exactly $6$ of the $8$ boxes are satisfactory.</div><p style="text-align: right;">[4]</p></li></ol>

National Fruit Company owns a large tomato farm. The tomatoes produced are harvested and sold in boxes of $25$. It is known that $100p\%$ of the tomatoes are rotten. For these boxes, the mean number of rotten tomatoes in a box is $1$.

Assume now that the context above is well-modelled by a binomial distribution.

Boxes that contains at least $24$ tomatoes that are not rotten are deemed satisfactory.

Binomial Models and Assumptions

Repeated Binomial Experiments

Determining an Unknown Binomial Success Probability

Mean and Variance of a Binomial Distribution

2021 NJC P2 Q8

<p>Amil and Babs are playing a game in which they take turns to throw a fair six-sided die. The winner is the first player to throw the same number as the other player has thrown immediately before. Anil throws first.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find exactly the probability that Anil wins the game.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given that Babs wins the game, find exactly the probability that she wins on her second throw.</div><p style="text-align: right;">[3]</p></li></ol>

Amil and Babs are playing a game in which they take turns to throw a fair six-sided die. The winner is the first player to throw the same number as the other player has thrown immediately before. Anil throws first.

N2022 P2 Q6

<p>A company sells toys in blind boxes. Each blind box contains exactly one toy. On average, $2\%$ of such blind boxes contain a faulty toy.</p>
<p>A collector buys $15$ such blind boxes at random.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, in context, two assumptions needed for the number of faulty toys he gets to be well modelled by a binomial distribution.</div><p style="text-align: right;">[2]</p></li></ol>
<p>Assume now that the number of faulty toys the collector gets has a binomial distribution.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the probability that the collector gets no less than $3$ faulty toys.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the probability that the fourteenth box is the second box with a faulty toy.</div><p style="text-align: right;">[2]</p></li></ol>
<p>The company also sells keychains. The probability of a keychain being faulty is $p$. The number of faulty keychains follows a binomial distribution. Faults on keychains are independent of faults on toys in blind boxes.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Write down an expression, in terms of $p$, for the probability that in a random sample of $3$ keychains, exactly one is faulty.</div><p style="text-align: right;">[1]</p></li></ol>
<p>A surprise pack contains $2$ randomly chosen blind boxes and $3$ randomly chosen keychains.</p>
<p>Given that a randomly selected surprise pack contains at most $1$ faulty item, the probability that one of the toys is the only faulty item in the pack is $0.035$.</p>
<ol class="i-list" start="5"><li><div class="list-item-paragraph">Write down an equation satisfied by $p$. Hence, find the value of $p$.</div><p style="text-align: right;">[4]</p></li></ol>

A company sells toys in blind boxes. Each blind box contains exactly one toy. On average, $2\%$ of such blind boxes contain a faulty toy.

A collector buys $15$ such blind boxes at random.

Assume now that the number of faulty toys the collector gets has a binomial distribution.

The company also sells keychains. The probability of a keychain being faulty is $p$. The number of faulty keychains follows a binomial distribution. Faults on keychains are independent of faults on toys in blind boxes.

A surprise pack contains $2$ randomly chosen blind boxes and $3$ randomly chosen keychains.

Given that a randomly selected surprise pack contains at most $1$ faulty item, the probability that one of the toys is the only faulty item in the pack is $0.035$.

Probabilities Involving Multiple Binomial Variables

2025 SAJC P2 Q8

<p>In a game of chance, a player has to spin a fair spinner. The spinner has $7$ sections and an arrow which has an equal chance of coming to rest over any of the $7$ sections. The spinner has $1$ section labelled $\mathbf{R}$, $2$ sections labelled $\mathbf{B}$ and $4$ sections labelled $\mathbf{Y}$ (see diagram).</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760955584642-1qpwfpsbvyoi.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA5NTU1ODQ2NDItMXFwd2Zwc2J2eW9pLnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTI3NDk2ODUsImV4cCI6MTc5Mjc0OTk4NX0.2RYdyIoU4LyKNqYVp_Y3cK4_qXHotIyXjV3_3omCKSI" width="400" class="aligncenter" /></figure>
<p>The player then has to throw one of three fair six-sided dice, coloured red, blue or yellow. If the spinner comes to rest over $\mathbf{R}$ the red die is thrown, if the spinner comes to rest over $\mathbf{B}$ the blue die is thrown and if the spinner comes to rest over $\mathbf{Y}$ the yellow die is thrown. The yellow die has one face with $*$ on it, the blue die has two faces with $*$ on it and the red die has three faces with $*$ on it. The player wins the game if the die thrown comes to rest with a face showing $*$ uppermost.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the probability that a player wins a game.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Given that a player wins a game, find the probability that the spinner came to rest over $B$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the probability that a player wins $3$ consecutive games, each time throwing a die of a different colour.</div><p style="text-align: right;">[2]</p></li></ol>

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

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

Math Question Bank