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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>In a group of $16$ students, $12$ take art and $8$ take music. One student takes neither art nor music. The Venn diagram below shows the events art and music. The values $p, q, r$ and $s$ represent numbers of students.</p>
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<ol class="abc-list"><li data-list-item-id="ibsl-q002-part-1-014" data-list-item-path="ibsl-q002-part-1-014"><div class="list-item-paragraph"><span class="inline-marks">[5]</span></div><ol class="i-list"><li data-list-item-id="ibsl-q002-part-1-1-015" data-list-item-path="ibsl-q002-part-1-014/ibsl-q002-part-1-1-015"><div class="list-item-paragraph"><p>Write down the value of $s$.</p></div></li><li data-list-item-id="ibsl-q002-part-1-2-016" data-list-item-path="ibsl-q002-part-1-014/ibsl-q002-part-1-2-016"><div class="list-item-paragraph"><p>Find the value of $q$.</p></div></li><li data-list-item-id="ibsl-q002-part-1-3-017" data-list-item-path="ibsl-q002-part-1-014/ibsl-q002-part-1-3-017"><div class="list-item-paragraph"><p>Write down the value of $p$ and of $r$.</p></div></li></ol></li><li data-list-item-id="ibsl-q002-part-2-018" data-list-item-path="ibsl-q002-part-2-018"><div class="list-item-paragraph"><span class="inline-marks">[4]</span></div><ol class="i-list"><li data-list-item-id="ibsl-q002-part-2-1-019" data-list-item-path="ibsl-q002-part-2-018/ibsl-q002-part-2-1-019"><div class="list-item-paragraph">A student is selected at random. Given that the student takes music, write down the probability the student takes art.</div></li><li data-list-item-id="ibsl-q002-part-2-2-020" data-list-item-path="ibsl-q002-part-2-018/ibsl-q002-part-2-2-020"><div class="list-item-paragraph">Hence, show that taking music and taking art are not independent events.</div></li></ol></li><li data-list-item-id="ibsl-q002-part-3-021" data-list-item-path="ibsl-q002-part-3-021"><div class="list-item-paragraph">Two students are selected at random, one after the other. Find the probability that the first student takes only music and the second student takes only art.<span class="inline-marks">[4]</span></div></li></ol>

In a group of $16$ students, $12$ take art and $8$ take music. One student takes neither art nor music. The Venn diagram below shows the events art and music. The values $p, q, r$ and $s$ represent numbers of students.

Two students are selected at random, one after the other. Find the probability that the first student takes only music and the second student takes only art.

Set Theory and Probability — Combined Problems

Independent Events

Conditional Probability

Dependent Events

Complete a Venn diagram, calculate conditional probability, test independence and sample without replacement.

Probability

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

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

Calculate ordered draws without replacement and show dependence.

<p>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.

Probability of a Single Event

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

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

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

Find the probability that the team wins the game.

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

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

<p>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.

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

<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.

Complementary Events

Applying Probability Laws

Probability Bounds and Inequalities

Addition Rule for Overlapping Events

<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 box contains $5$ red balls, $5$ green balls and $5$ blue balls. The balls are identical in size. Balls of the same colour are each printed with a distinct number from $1$, $2$, $3$, $4$, $5$. Two balls are randomly drawn without replacement. Find the probability that</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">at least $1$ red ball is drawn,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the sum of the numbers on the balls drawn is at least $9$,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the sum of the numbers on the balls drawn is at least $9$ given that at least $1$ red ball is drawn.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>State, with a reason, whether the events 'the sum of the numbers on the balls drawn is at least $9$' and 'at least $1$ red ball is drawn' are independent.<span class="inline-marks">[1]</span></p>

A box contains $5$ red balls, $5$ green balls and $5$ blue balls. The balls are identical in size. Balls of the same colour are each printed with a distinct number from $1$, $2$, $3$, $4$, $5$. Two balls are randomly drawn without replacement. Find the probability that

the sum of the numbers on the balls drawn is at least $9$,

the sum of the numbers on the balls drawn is at least $9$ given that at least $1$ red ball is drawn.

State, with a reason, whether the events 'the sum of the numbers on the balls drawn is at least $9$' and 'at least $1$ red ball is drawn' are independent.

Selections with Restrictions

Probability Using Permutations and Combinations

Complementary Counting

2013 DHS P2 Q7

Permutation and Combination

<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

2013 TJC P2 Q10

<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 Probabilities Using Technology

Binomial Models and Assumptions

Binomial Probabilities Using the Probability Mass Function

Repeated Binomial Experiments

Determining an Unknown Binomial Success Probability

Mean and Variance of a Binomial Distribution

2021 NJC P2 Q8

Discrete Random Variables (DRV)

<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><strong>In this question you should state the parameters of any distributions you use.</strong></p>
<p>At a burger shop, the wait time, $W$ (in minutes), is defined to be the time from when a customer places an order at the counter until the food is collected. It was proposed that $W$ is modelled by $N(2,{{1.5}^{2}})$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Give a reason why this model is not suitable.</div><p style="text-align: right;">[1]</p></li></ol>
<p>A new model for $W$ is given by $N(5,{{1}^{2}})$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find the range of values of $k$ such that at least $90\%$ of customers experience a wait time longer than $k$ minutes.</div><p style="text-align: right;">[2]</p></li></ol>
<p>A customer bought burgers from the shop on three independent occasions, with wait times denoted by ${{W}_{1}}$, ${{W}_{2}}$ and ${{W}_{3}}$ respectively. Let $\bar{W}=\frac{{{W}_{1}}+{{W}_{2}}+{{W}_{3}}}{3}$.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Find the values of $\mathrm{Var}(\bar{W}-{{W}_{1}})$ and $\mathrm{Var}(\bar{W})+\mathrm{Var}({{W}_{1}})$ and hence, determine whether $\mathrm{Var}(\bar{W}-{{W}_{1}})=\mathrm{Var}(\bar{W})+\mathrm{Var}({{W}_{1}})$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the probability that the mean wait time is within one minute of the wait time on the first occasion.</div><p style="text-align: right;">[3]</p></li></ol>
<p>On the 4th occasion, the customer went to the restroom immediately after the order was placed. The time spent in the restroom, in $T$ minutes, is modelled by $T\sim N(7,{{1.5}^{2}})$.</p>
<p>Assume that the time taken to walk to the restroom is negligible and $W$ and $T$ follow independent normal distributions.</p>
<ol class="abc-list" start="5"><li><div class="list-item-paragraph">The customer is in a rush and will not wait for more than 3 minutes after leaving the restroom. Given that the burger is not ready for collection after the customer leaves the restroom, find the probability that the customer will leave the shop without collecting the burger.</div><p style="text-align: right;">[4]</p></li></ol>
<p>To reduce the wait time for customers, the shop later installs self-order kiosks. The wait time from when a customer places an order at the kiosk until the food is collected is modelled as being reduced by $20\%$ compared to that at the counter.</p>
<ol class="abc-list" start="6"><li><div class="list-item-paragraph">8 customers who ordered at the counter and 8 customers who ordered at the kiosk are randomly selected and their wait times are recorded. The wait time of each customer is independent of the others. Find the probability that exactly 2 of these 16 customers have a wait time of at least 5 minutes.</div><p style="text-align: right;">[4]</p></li></ol>

**In this question you should state the parameters of any distributions you use.**

At a burger shop, the wait time, $W$ (in minutes), is defined to be the time from when a customer places an order at the counter until the food is collected. It was proposed that $W$ is modelled by $N(2,{{1.5}^{2}})$.

A new model for $W$ is given by $N(5,{{1}^{2}})$.

A customer bought burgers from the shop on three independent occasions, with wait times denoted by ${{W}_{1}}$, ${{W}_{2}}$ and ${{W}_{3}}$ respectively. Let $\bar{W}=\frac{{{W}_{1}}+{{W}_{2}}+{{W}_{3}}}{3}$.

On the 4th occasion, the customer went to the restroom immediately after the order was placed. The time spent in the restroom, in $T$ minutes, is modelled by $T\sim N(7,{{1.5}^{2}})$.

Assume that the time taken to walk to the restroom is negligible and $W$ and $T$ follow independent normal distributions.

To reduce the wait time for customers, the shop later installs self-order kiosks. The wait time from when a customer places an order at the kiosk until the food is collected is modelled as being reduced by $20\%$ compared to that at the counter.

Normal Probabilities and Quantiles

Probabilities Involving Multiple Binomial Variables

Assumptions for Sampling

Linear Transformations of a Normal Variable

Normal Models for Sample Means and Averages

Linear Combinations of Independent Normal Variables

Assessing the Suitability of a Normal Model

Sample Mean and Its Sampling Distribution

2025 HCI P2 Q10

Continuous Random Variables

Sampling

<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 $0.035$.

2025 SAJC P2 Q8

<p><strong>In this question, you should state the parameters of any distribution you use.</strong></p>
<p>A ceramic shop sells handmade ceramic cups. The mouths of the cups are assumed to be circular in shape. The diameters of the outer circumferences of the top rim of the cups, $S$, are assumed to follow a normal distribution with mean $\mu $ mm and standard deviation $\sigma $ mm.</p>
<ol class="abc-list"><li data-list-item-id="item-1760848344195-i903htzgu" data-list-item-path="item-1760848344195-i903htzgu"><div class="list-item-paragraph">It is given that&nbsp;$\mathrm{P}\!\left( S \lt 80.5 \right) = \mathrm{P}\!\left( S \gt 84.5 \right)$ and that the probability of the diameter of the outer circumference of the top rim of a randomly chosen cup being more than $85$mm is $1.15\%$. Find the value of $\mu $, and show that&nbsp;$\sigma =1.10$, when corrected to $3$ significant figures.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760848345344-f2hyo0hpc" data-list-item-path="item-1760848345344-f2hyo0hpc"><div class="list-item-paragraph">The shop also makes covers of circular shape that can be fitted over the mouths of the cups. The diameter of any randomly chosen cover, $C$, in mm, follows a normal distribution with mean $83$ mm and standard deviation $1.5$mm. A cover would be considered to be well-fitted over the mouth of a cup if the diameter of the cover is larger than that of the outer circumference of the top rim of the cup by not more than $2$ mm. Find the probability that a randomly chosen cover is well-fitted over the mouth of a randomly chosen cup.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760848346573-zbviy88fi" data-list-item-path="item-1760848346573-zbviy88fi"><div class="list-item-paragraph">A cover and a cup are randomly chosen. If the cover is well-fitted over the mouth of the cup, find the probability that the diameter of the cover is larger than that of the cup by more than $1.5$ mm.<span class="inline-marks">[3]</span></div></li></ol>

**In this question, you should state the parameters of any distribution you use.**

A ceramic shop sells handmade ceramic cups. The mouths of the cups are assumed to be circular in shape. The diameters of the outer circumferences of the top rim of the cups, $S$, are assumed to follow a normal distribution with mean $\mu $ mm and standard deviation $\sigma $ mm.

Finding Unknown Normal Mean and Standard Deviation

2024 HCI P2 Q9

<p>A computer randomly chooses $10$ shapes from $18$ squares and $12$ triangles. The number of squares chosen is denoted by $S$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760836044009-pb3skgh65" data-list-item-path="item-1760836044009-pb3skgh65"><div class="list-item-paragraph">The most probable number of squares that the computer chooses is denoted by $s$. By using the fact that&nbsp;$P\left( S=s \right)&gt;P\left( S=s+1 \right)$, show that&nbsp;$s$ satisfies the inequality</div><div class="list-item-paragraph" style="text-align: center">$\left( s+1 \right)\left( s+3 \right)&gt;\left( a-s \right)\left( b-s \right)$</div><div class="list-item-paragraph">where&nbsp;$a$ and $b$ are constants to be determined. Hence find the value of $s$.<span class="inline-marks">[4]</span></div></li></ol>
<p>In a game, the computer randomly chooses $6$ squares and $4$ triangles. The squares are numbered from $1$ to $6$ and the triangles are numbered from $1$ to $4$. A square and a triangle are randomly chosen. Let&nbsp;$X$ be the score which is defined as the difference (taken as always positive) between the numbers of the chosen square and triangle.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760836072225-u40wni23o" data-list-item-path="item-1760836072225-u40wni23o"><div class="list-item-paragraph">Find the probability distribution of $X$.<span class="inline-marks">[3]</span></div></li></ol>
<p>Three independent games are played. The total score of the games is $3$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-1760836085229-km6pguiff" data-list-item-path="item-1760836085229-km6pguiff"><div class="list-item-paragraph">Find the probability that the score of the first game is $2$.<span class="inline-marks">[3]</span></div></li></ol>

A computer randomly chooses $10$ shapes from $18$ squares and $12$ triangles. The number of squares chosen is denoted by $S$.

In a game, the computer randomly chooses $6$ squares and $4$ triangles. The squares are numbered from $1$ to $6$ and the triangles are numbered from $1$ to $4$. A square and a triangle are randomly chosen. Let $X$ be the score which is defined as the difference (taken as always positive) between the numbers of the chosen square and triangle.

Three independent games are played. The total score of the games is $3$.

Constructing and Using Discrete Probability Distributions

Linear Combinations of Random Variables

2024 NYJC P2 Q7

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

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

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

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

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

Law of Total Probability

2024 SAJC P2 Q5

<p>For events $A$ and $B$, it is given that&nbsp;$P\left( A \right)=\frac{1}{8}$, $P\left( A\cup B \right)=\frac{9}{16}$ and $P\left( A\cap {B}' \right)=\frac{1}{16}$.</p>
<ol class="i-list"><li data-list-item-id="item-1760781172283-qq04lwcra" data-list-item-path="item-1760781172283-qq04lwcra"><div class="list-item-paragraph">Find $P\left( B \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760781174061-kzrl5fl7k" data-list-item-path="item-1760781174061-kzrl5fl7k"><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;$P\left( C \right)=\frac{3}{8}$, $P\left( A\cap B\cap C \right)=\frac{1}{64}$ and $C$ is independent of $A$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760781194927-07nr12r5q" data-list-item-path="item-1760781194927-07nr12r5q"><div class="list-item-paragraph">Find $P\left( {A}'\cap C \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760781195819-twm664oux" data-list-item-path="item-1760781195819-twm664oux"><div class="list-item-paragraph">Find the range of values for&nbsp;$P\left( A'\cap {B}'\cap {C}' \right)$.<span class="inline-marks">[3]</span></div></li></ol>

For events $A$ and $B$, it is given that $P\left( A \right)=\frac{1}{8}$, $P\left( A\cup B \right)=\frac{9}{16}$ and $P\left( A\cap {B}' \right)=\frac{1}{16}$.

For a third event $C$, it is given that $P\left( C \right)=\frac{3}{8}$, $P\left( A\cap B\cap C \right)=\frac{1}{64}$ and $C$ is independent of $A$.

Mutually Exclusive Events

2024 TMJC P2 Q5

<p>The events $A$ and $B$ are such that&nbsp;$\mathrm{P}\left( A \right)=a$ and $\mathrm{P}\left( B \right)=b$. $A$ and $B$ are independent events.</p>
<ol class="abc-list"><li data-list-item-id="item-1760781012593-etiznm953" data-list-item-path="item-1760781012593-etiznm953"><div class="list-item-paragraph">Find an expression for&nbsp;$\mathrm{P}\left( {A}'\cap {B}' \right)$ in terms of $a$ and $b$, and hence prove that&nbsp;${A}'$ and ${B}'$ are independent events.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is given that&nbsp;$\mathrm{P}\left( A\mid {B}' \right)=0.85$ and ${\mathrm{P}\left( {{B}'} \right)=0.8}$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760781026636-y4nwlf8ee" data-list-item-path="item-1760781026636-y4nwlf8ee"><div class="list-item-paragraph">Find $\mathrm{P}\left( A\cap {B}' \right)$.<span class="inline-marks">[2]</span></div></li></ol>
<p>For a third event $C$, it is given that&nbsp;$A$ and $C$ are mutually exclusive and $\mathrm{P}\left( {A}'\cap {C}' \right)=0.52$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-1760781041671-n28p3vsy0" data-list-item-path="item-1760781041671-n28p3vsy0"><div class="list-item-paragraph">Find $\mathrm{P}\left( C \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760781044055-x873mmx49" data-list-item-path="item-1760781044055-x873mmx49"><div class="list-item-paragraph">Hence find the set of possible values of $\mathrm{P}\left( {A}'\cap {B}'\cap {C}' \right)$.<span class="inline-marks">[3]</span></div></li></ol>

The events $A$ and $B$ are such that $\mathrm{P}\left( A \right)=a$ and $\mathrm{P}\left( B \right)=b$. $A$ and $B$ are independent events.

It is given that $\mathrm{P}\left( A\mid {B}' \right)=0.85$ and ${\mathrm{P}\left( {{B}'} \right)=0.8}$.

For a third event $C$, it is given that $A$ and $C$ are mutually exclusive and $\mathrm{P}\left( {A}'\cap {C}' \right)=0.52$.

2024 ACJC P2 Q6

<p>There are $36$ people at a gathering of two families. There are $25$ people with the name Lee and $11$ people with the name Chan. Of the $25$ people named Lee, $4$ are single men, $5$ are single women and there are $8$ married couples. Of the $11$ people named Chan, $2$ are single men, $3$ are single women and there are $3$ married couples. Two people are chosen at random from the gathering.</p>
<ol class="abc-list"><li data-list-item-id="item-1760776839984-pcjbp5t8u" data-list-item-path="item-1760776839984-pcjbp5t8u"><div class="list-item-paragraph">Show that the probability that they both have the name Lee is $\frac{10}{21}$.</div></li><li data-list-item-id="item-1760776840629-42q8pvsux" data-list-item-path="item-1760776840629-42q8pvsux"><div class="list-item-paragraph">Find the probability that they are married to each other.</div></li><li data-list-item-id="item-1760776841544-v176xlmfi" data-list-item-path="item-1760776841544-v176xlmfi"><div class="list-item-paragraph">Find the probability that they both have the name Lee, given that they are married to each other.</div></li><li data-list-item-id="item-1760776842931-tjq7p1y7c" data-list-item-path="item-1760776842931-tjq7p1y7c"><div class="list-item-paragraph">Find the probability that they are a man and a woman with the same name.</div></li><li data-list-item-id="item-1760776844094-y1cmq6zah" data-list-item-path="item-1760776844094-y1cmq6zah"><div class="list-item-paragraph">Find the probability that they are married to each other, given that they are a man and a woman with the same name.</div></li></ol>

There are $36$ people at a gathering of two families. There are $25$ people with the name Lee and $11$ people with the name Chan. Of the $25$ people named Lee, $4$ are single men, $5$ are single women and there are $8$ married couples. Of the $11$ people named Chan, $2$ are single men, $3$ are single women and there are $3$ married couples. Two people are chosen at random from the gathering.

N1997 P2 Q6

<p>Four rods of lengths $1$, $2$, $3$ and $4$ units are placed in a bag from which one rod is selected at random. The probability of selecting a rod of length $l$ is $kl$. Find the value of $k$.</p>
<p>Show that the expected value of $X$, the length of the selected rod, is $3$ units and find the variance of $X$.</p>
<p>After a rod has been selected it is not replaced. The probabilities of selection for each of the three rods remain are in the same ratio as they were before the first selection. A second rod is now selected from the bag. Defining $Y$ to be the length of the rod and writing ${{P}_{1}}=\mathrm{P}\left( Y=1|X=2 \right)$, ${{P}_{2}}=\mathrm{P}\left( Y=2|X=1 \right)$ show that $16{{P}_{1}}=9{{P}_{2}}$.</p>
<p>Show also that $\mathrm{P}\left( X+Y=3 \right)=\frac{17}{360}$.</p>

Four rods of lengths $1$, $2$, $3$ and $4$ units are placed in a bag from which one rod is selected at random. The probability of selecting a rod of length $l$ is $kl$. Find the value of $k$.

Show that the expected value of $X$, the length of the selected rod, is $3$ units and find the variance of $X$.

After a rod has been selected it is not replaced. The probabilities of selection for each of the three rods remain are in the same ratio as they were before the first selection. A second rod is now selected from the bag. Defining $Y$ to be the length of the rod and writing ${{P}_{1}}=\mathrm{P}\left( Y=1|X=2 \right)$, ${{P}_{2}}=\mathrm{P}\left( Y=2|X=1 \right)$ show that $16{{P}_{1}}=9{{P}_{2}}$.

Show also that $\mathrm{P}\left( X+Y=3 \right)=\frac{17}{360}$.

Math Question Bank