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>A company buys $p\%$ of its electronic components from supplier $A$ and the remaining $(100 - p)\%$ from supplier $B$. The probability that a randomly chosen component supplied by $A$ is faulty is $0.05$. The probability that a randomly chosen component supplied by $B$ is faulty is $0.03$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$p = 25$, find the probability that a randomly chosen component is faulty.</span></div></li><li><div class="list-item-paragraph">For a general value of $p$, the probability that a randomly chosen component that is faulty was supplied by $A$ is denoted by $\mathrm{f}(p)$. Show that&nbsp;$\mathrm{f}(p) = \frac{0.05p}{0.02p + 3}$. Prove by differentiation that&nbsp;$\mathrm{f}$ is an increasing function for&nbsp;$0 \leq p \leq 100$, and explain what this statement means in the context of the question.</div></li></ol>

A company buys $p\%$ of its electronic components from supplier $A$ and the remaining $(100 - p)\%$ from supplier $B$. The probability that a randomly chosen component supplied by $A$ is faulty is $0.05$. The probability that a randomly chosen component supplied by $B$ is faulty is $0.03$.

Given that $p = 25$, find the probability that a randomly chosen component is faulty.

Bayes' Theorem

Quotient Rule

Law of Total Probability

Increasing and Decreasing Functions

N2009 P2 Q7

Techniques of Differentiation

Probability

Applications of Differentiation

<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

Complementary Events

Conditional Probability

Probability Using Permutations and Combinations

Constructing and Using Probability Trees

Conditional Probability from Tree Diagrams

2013 TJC P2 Q10

<p>In a hypothetical population, $2\%$ have a disease. A test returns positive for&nbsp;$90\%$ of those with the disease and negative for&nbsp;$90\%$ of those without it. A person is selected uniformly at random from the population.</p>
<ol class="i-list"><li data-list-item-id="item-1766237404568-r3fvjtm8z" data-list-item-path="item-1766237404568-r3fvjtm8z"><div class="list-item-paragraph">Find the probability that a person tests positive.</div></li><li data-list-item-id="item-1766237405745-fzy69zbqi" data-list-item-path="item-1766237405745-fzy69zbqi"><div class="list-item-paragraph">It is given that person tests positive. What is the chance they actually have the disease?</div></li></ol>

In a hypothetical population, $2\%$ have a disease. A test returns positive for $90\%$ of those with the disease and negative for $90\%$ of those without it. A person is selected uniformly at random from the population.

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA5NTU1ODQ2NDItMXFwd2Zwc2J2eW9pLnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTI3NDk3MDgsImV4cCI6MTc5Mjc1MDAwOH0.NR5J7Sbn8c_j7PmmyZnAW3R8RBB3GKbQvN4ahgH_xOE" 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.

Probabilities of Successive Events

N2016 P2 Q5

<p>A company sells consumer hobby drones, and they buy the rotor motor parts from two suppliers, AE Rotors and Brushless Inc. ${25\%}$ of the motors are from AE Rotors and the rest are from Brushless Inc. The probability that a randomly chosen motor supplied by AE Rotors is faulty is $\frac{1}{20}$. The probability that a randomly chosen motor supplied by Brushless Inc is faulty is $q$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760781269010-ow7oymcdt" data-list-item-path="item-1760781269010-ow7oymcdt"><div class="list-item-paragraph">Complete the tree diagram below to illustrate the possibilities for a randomly chosen motor. You should use the following notations for the events:</div><div class="list-item-paragraph" style="text-align: center"><p style="text-align: left;">$A$ for the motor is supplied by AE Rotors. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $B$ for the motor is supplied by Brushless Inc.</p></div><div class="list-item-paragraph" style="text-align: center"><p style="text-align: left;">$F$ for the motor is faulty, and &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $\quad\,{F}'$ for its complement, i.e. the motor is not faulty.</p><span class="inline-marks">[1]</span></div></li></ol>
<p>It is known that the probability that a randomly picked motor that is faulty is supplied by Brushless Inc is $\frac{3}{8}$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760781281010-gszqiiyg5" data-list-item-path="item-1760781281010-gszqiiyg5"><div class="list-item-paragraph">Use the above information to set up an equation involving $q$, and hence find the value of $q$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760781281843-6x9qpzzhr" data-list-item-path="item-1760781281843-6x9qpzzhr"><div class="list-item-paragraph">Three motors are randomly picked. Find the probability that all of them are supplied by AE Rotors and exactly one of them is faulty.<span class="inline-marks">[2]</span></div></li></ol>

A company sells consumer hobby drones, and they buy the rotor motor parts from two suppliers, AE Rotors and Brushless Inc. ${25\%}$ of the motors are from AE Rotors and the rest are from Brushless Inc. The probability that a randomly chosen motor supplied by AE Rotors is faulty is $\frac{1}{20}$. The probability that a randomly chosen motor supplied by Brushless Inc is faulty is $q$.

It is known that the probability that a randomly picked motor that is faulty is supplied by Brushless Inc is $\frac{3}{8}$.

Finding Unknowns in Probability Models

Binomial Probabilities Using the Probability Mass Function

2024 EJC P2 Q5

Discrete Random Variables (DRV)

<p>$16\%$ of all phone calls are scam calls. A company designed a $2$-stage filtering anti-scam app, ${{X}^{2}}$scam, that can block scam calls which plague mobile phone users. In a pre-launch test, it is found that ${{X}^{2}}$scam blocked $88\%$ of calls that are scam calls and $5\%$ of calls that are non-scam calls in the first stage of filtering. Of the calls that are unblocked in the first stage, ${{X}^{2}}$scam blocks $99\%$of scam calls and $5\%$ of non-scam calls within the second stage. Calls that are blocked in the first stage do not go through a second stage of filtering.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Draw a tree diagram to represent the above information.</div><p style="text-align: right;">[2]</p></li></ol>
<p>A call is selected at random. Find the probability that</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">it is a scam call that is blocked.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">it is a scam call, given that the call is blocked.</div><p style="text-align: right;">[2]</p></li></ol>
<p>Three calls were selected at random.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find the probability that ${{X}^{2}}$scam correctly filters exactly two of the calls.</div><p style="text-align: right;">[4]</p></li></ol>

$16\%$ of all phone calls are scam calls. A company designed a $2$-stage filtering anti-scam app, ${{X}^{2}}$scam, that can block scam calls which plague mobile phone users. In a pre-launch test, it is found that ${{X}^{2}}$scam blocked $88\%$ of calls that are scam calls and $5\%$ of calls that are non-scam calls in the first stage of filtering. Of the calls that are unblocked in the first stage, ${{X}^{2}}$scam blocks $99\%$of scam calls and $5\%$ of non-scam calls within the second stage. Calls that are blocked in the first stage do not go through a second stage of filtering.

A call is selected at random. Find the probability that

2022 EJC J2 MYE P2 Q9

<p>In this question you should state clearly all the distributions that you use, together with the values of the appropriate parameters. You should also assume that&nbsp;$T$ and $X$ follow independent normal distributions.</p>
<p>Each KTX train takes $T$ minutes to travel from Seoul Train Station to Pohang Train Station. It is known that&nbsp;$T$ follows the distribution $\mathrm{N}\left( 144,25 \right)$.</p>
<ol class="i-list"><li data-list-item-id="item-1760683995445-7hkeiz5xq" data-list-item-path="item-1760683995445-7hkeiz5xq"><div class="list-item-paragraph">The probability that a randomly selected KTX train takes more than $k$ hours to reach Pohang is $0.5$. Without the use of a calculator, explain why $k=2.4$.<span class="inline-marks">[1]</span></div></li></ol>
<p>Before 7pm daily, each express bus takes $X$ minutes to travel from Seoul Express Bus Terminal to Pohang Express Bus Terminal. It is known that&nbsp;$X$ follows the distribution $\mathrm{N}\left( 236,81 \right)$. After 7pm daily, the travel time taken by each express bus will be reduced by $10\%$ as all the express buses will not make a stop at Daegu.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1760684011068-dbw9y30cu" data-list-item-path="item-1760684011068-dbw9y30cu"><div class="list-item-paragraph">Find the probability that after 7pm, to travel from Seoul to Pohang, the travel time of a randomly selected KTX train is at most an hour faster than the travel time of a randomly selected express bus.<span class="inline-marks">[4]</span></div></li></ol>
<p>Kim needs to travel from Seoul to Pohang on Saturday mornings to visit his parents. He prefers to take a KTX train if the tickets are available. On average, $70\%$ of his journeys are by train.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760684029031-t73scwwhd" data-list-item-path="item-1760684029031-t73scwwhd"><div class="list-item-paragraph">On a particular Saturday morning, there was a train delay of $1.6$ hours due to a train fault. Given that Kim took more than $4$ hours to reach Pohang on that morning, find the probability that Kim travels to Pohang by train.<span class="inline-marks">[3]</span></div></li></ol>
<p>The cost of taking a KTX train and taking an express bus from Seoul to Pohang are $\$54$ and $\$24$ respectively.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1760684049850-pdiije9gr" data-list-item-path="item-1760684049850-pdiije9gr"><div class="list-item-paragraph">Find Kim’s expected cost of travelling (one-way) from Seoul to Pohang.<span class="inline-marks">[1]</span></div></li></ol>

In this question you should state clearly all the distributions that you use, together with the values of the appropriate parameters. You should also assume that $T$ and $X$ follow independent normal distributions.

Each KTX train takes $T$ minutes to travel from Seoul Train Station to Pohang Train Station. It is known that $T$ follows the distribution $\mathrm{N}\left( 144,25 \right)$.

Before 7pm daily, each express bus takes $X$ minutes to travel from Seoul Express Bus Terminal to Pohang Express Bus Terminal. It is known that $X$ follows the distribution $\mathrm{N}\left( 236,81 \right)$. After 7pm daily, the travel time taken by each express bus will be reduced by $10\%$ as all the express buses will not make a stop at Daegu.

Kim needs to travel from Seoul to Pohang on Saturday mornings to visit his parents. He prefers to take a KTX train if the tickets are available. On average, $70\%$ of his journeys are by train.

The cost of taking a KTX train and taking an express bus from Seoul to Pohang are $\$54$ and $\$24$ respectively.

Normal Probabilities and Quantiles

Symmetry and Properties of the Normal Distribution

Linear Transformations of a Normal Variable

Linear Combinations of Independent Normal Variables

2022 ACJC P2 Q9

Continuous Random Variables

<p>It is known that in a certain country, $30\%$ of the people support Party A; $50\%$ of the people support Party B; and $20\%$ of the people support Party C.</p>
<p>In the recent election, $55\%$ of Party A supporters voted, as did $86\%$ of the supporters of Party B and $42\%$ of the supporters of Party C.</p>
<ol class="abc-list"><li data-list-item-id="item-1760434394931-tarcz50ty" data-list-item-path="item-1760434394931-tarcz50ty"><div class="list-item-paragraph">One person is selected at random. Find the probability that</div><ol class="i-list"><li data-list-item-id="subitem-1760434405739-kzhw65rfc" data-list-item-path="item-1760434394931-tarcz50ty/subitem-1760434405739-kzhw65rfc"><div class="list-item-paragraph">the person voted,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760434412008-t3wfs4g6h" data-list-item-path="item-1760434394931-tarcz50ty/subitem-1760434412008-t3wfs4g6h"><div class="list-item-paragraph">the person is a supporter of Party B, given that the person voted.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-1760434395741-gus7rlq94" data-list-item-path="item-1760434395741-gus7rlq94"><div class="list-item-paragraph">Two persons are selected at random. Find the probability that exactly one of them voted.<span class="inline-marks">[2]</span></div></li></ol>

It is known that in a certain country, $30\%$ of the people support Party A; $50\%$ of the people support Party B; and $20\%$ of the people support Party C.

In the recent election, $55\%$ of Party A supporters voted, as did $86\%$ of the supporters of Party B and $42\%$ of the supporters of Party C.

2023 VJC P2 Q6

<p>Diabetes is present in $27\%$ of the elderly population. In a health screening exercise, a quick test for diabetes is used, but the test is not totally reliable. It has&nbsp;$0.98$ probability of giving a positive result when an elderly has diabetes, and $0.04$ probability of giving a positive result when an elderly does not have diabetes. A randomly chosen elderly is given the test. Find the probability that</p>
<ol class="i-list"><li data-list-item-id="item-1760431941810-k5ultf6kk" data-list-item-path="item-1760431941810-k5ultf6kk"><div class="list-item-paragraph">the result of the test is positive,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760431942403-xdfkx3fbm" data-list-item-path="item-1760431942403-xdfkx3fbm"><div class="list-item-paragraph">the elderly has diabetes given that the result of the test is positive.<span class="inline-marks">[2]</span></div></li></ol>

Diabetes is present in $27\%$ of the elderly population. In a health screening exercise, a quick test for diabetes is used, but the test is not totally reliable. It has $0.98$ probability of giving a positive result when an elderly has diabetes, and $0.04$ probability of giving a positive result when an elderly does not have diabetes. A randomly chosen elderly is given the test. Find the probability that

2013 IJC P2 Q5

<p>A certain disease is present in $1$ in $200$ of the population. In a mass screening programme, a quick test for the disease is used but the test is not totally reliable. For someone who does have the disease there is a probability of $0.9$ that the test will prove positive, whereas for someone who does not have the disease, there is a probability of $0.02$ that the test will prove positive.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">A person is chosen at random and tested.</div><div class="list-item-image"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760417060035-89gldovncl5.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA0MTcwNjAwMzUtODlnbGRvdm5jbDUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc0OTcwOCwiZXhwIjoxNzkyNzUwMDA4fQ.Rod6411acX7dSkIFcdPqNnLBrZ-DrZncTmKC7ndh0bU" alt="List item image" /></div><ol class="i-list"><li><div class="list-item-paragraph">Copy and complete the tree diagram which illustrates the application of one test.</div></li><li><div class="list-item-paragraph">Find the probability that the person has the disease and the test is positive.</div></li><li><div class="list-item-paragraph">Find the probability that the test is negative.</div></li><li><div class="list-item-paragraph">Given that the test is positive, find the probability that the person has the disease.</div></li></ol></li><li><div class="list-item-paragraph">People for whom the test proves positive are recalled and re-tested. Find the probability that a person has the disease if the second test also proves positive.</div></li></ol>

A certain disease is present in $1$ in $200$ of the population. In a mass screening programme, a quick test for the disease is used but the test is not totally reliable. For someone who does have the disease there is a probability of $0.9$ that the test will prove positive, whereas for someone who does not have the disease, there is a probability of $0.02$ that the test will prove positive.

N1997 P2 Q3

<p>On average $8\%$ of a certain brand of kitchen lights are faulty. The lights are sold in boxes of $12$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, in context, two assumptions needed for the number of faulty lights in a box 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 lights in a box has a binomial distribution.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the probability that a box of $12$ of these kitchen lights contains at least $1$ faulty lights.</div><p style="text-align: right;">[1]</p></li></ol>
<p>The boxes are packed into cartons. Each carton contains $20$ boxes.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the probability that each box in one randomly selected carton contains at least one faulty light.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the probability that there are at least $20$ faulty lights in a randomly selected carton.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Explain why the answer to part <strong>(iv)</strong> is greater than the answer to part <strong>(iii)</strong>.</div><p style="text-align: right;">[1]</p></li></ol>
<p>The manufacturer introduces a quick test to check if lights are faulty. Lights identified as faulty are discarded. If a light is faulty there is a $95\%$ chance that the quick test will correctly identify the light as faulty. If the light is not faulty, there is a $6\%$ chance that the quick test will incorrectly identify the light as faulty.</p>
<ol class="i-list" start="6"><li><div class="list-item-paragraph">Find the probability that a light identified as faulty by the quick test is not faulty.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the probability that the quick test correctly identifies lights as faulty or not faulty.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Discuss briefly whether the quick test is worthwhile.</div><p style="text-align: right;">[1]</p></li></ol>

On average $8\%$ of a certain brand of kitchen lights are faulty. The lights are sold in boxes of $12$.

Assume now that the number of faulty lights in a box has a binomial distribution.

The boxes are packed into cartons. Each carton contains $20$ boxes.

The manufacturer introduces a quick test to check if lights are faulty. Lights identified as faulty are discarded. If a light is faulty there is a $95\%$ chance that the quick test will correctly identify the light as faulty. If the light is not faulty, there is a $6\%$ chance that the quick test will incorrectly identify the light as faulty.

Binomial Probabilities Using Technology

Binomial Models and Assumptions

Repeated Binomial Experiments

Multistage Binomial Sampling

N2017 P2 Q9

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

In a game, two boxes have the following contents.

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

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

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

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

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

Dependent Events

2022 SAJC P2 Q8

<p>The probability of obtaining a ‘$6$’ on a biased cubical dice is thrice the probability of rolling every other number on the dice.</p>
<p>Show that the probability of obtaining a ‘$6$’ on the biased dice is $\frac{3}{8}$.<span class="inline-marks">[1]</span></p>
<p>This biased dice is put into a bag together with $3$ fair cubical dice.</p>
<ol class="i-list"><li data-list-item-id="item-1759814610426-yojbikwur" data-list-item-path="item-1759814610426-yojbikwur"><div class="list-item-paragraph">One of the dice is chosen randomly from the bag and rolled once. Find the probability of obtaining a ‘$6$’.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759814611003-c52ba1y7e" data-list-item-path="item-1759814611003-c52ba1y7e"><div class="list-item-paragraph">One of the dice is chosen randomly and is rolled $n$ times.</div><ol class="abc-list"><li data-list-item-id="subitem-1759814614506-ey4h222i0" data-list-item-path="item-1759814611003-c52ba1y7e/subitem-1759814614506-ey4h222i0"><div class="list-item-paragraph">Find the probability that ‘$6$’ is obtained on all the $n$ rolls.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1759814615564-939f0rvnc" data-list-item-path="item-1759814611003-c52ba1y7e/subitem-1759814615564-939f0rvnc"><div class="list-item-paragraph">Given that ‘$6$’ is obtained for all the $n$ rolls, the probability that the biased dice is chosen is more than $0.95$. By forming an inequality in terms of $n$, solve for the least value of $n$.<span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-1759814611933-fqehhktyk" data-list-item-path="item-1759814611933-fqehhktyk"><div class="list-item-paragraph">One of the dice is chosen randomly from the bag and rolled once. The dice is then put back into the bag. Another dice is chosen randomly from the bag and rolled once. For every ‘$6$’ obtained, the player gains $\$2$, otherwise the player loses</div><div class="list-item-paragraph">$\$k$. If the game is fair, that is, expected winnings of the player is $\$0$, show that the value of $k$ is $0.56$.<span class="inline-marks">[3]</span></div><div class="list-item-paragraph">Find the variance of the player’s winnings.<span class="inline-marks">[2]</span></div></li></ol>

The probability of obtaining a ‘$6$’ on a biased cubical dice is thrice the probability of rolling every other number on the dice.

Show that the probability of obtaining a ‘$6$’ on the biased dice is $\frac{3}{8}$.

This biased dice is put into a bag together with $3$ fair cubical dice.

Expectation and Variance of Discrete Random Variables

Determining Unknowns in Discrete Probability Distributions

Expected Payoffs and Fair Games

Linear Combinations of Random Variables

2018 VJC P2 Q8

<p>A student has requested for a testimonial to apply for a job.</p>
<p>She estimates that there is an 80% chance of getting the job if she receives an excellent testimonial, a 40% chance if she receives a moderately good testimonial and a 10% chance if she receives a fair testimonial.</p>
<p>She further estimates that the probabilities that the testimonial will be excellent, moderate or fair are 0.7, 0.2 and 0.1 respectively.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Draw a tree diagram to illustrate the information and calculate the exact probability of the students getting the job.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given that she receives the job offer, what is the exact probability that she received an excellent testimonial?<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given that she did not receive a moderate testimonial, what is the exact probability that she did not receive a job offer?<span class="inline-marks">[2]</span></div></li></ol>

A student has requested for a testimonial to apply for a job.

She estimates that there is an 80% chance of getting the job if she receives an excellent testimonial, a 40% chance if she receives a moderately good testimonial and a 10% chance if she receives a fair testimonial.

She further estimates that the probabilities that the testimonial will be excellent, moderate or fair are 0.7, 0.2 and 0.1 respectively.

2008 SAJC P2 Q8

<p>I have a choice of two routes to get to school. The probability that I am punctual for school is $\frac{43}{50}$ and the probability that I choose the first route is $\frac{3}{5}$ . If I get to school punctually, the probability that the first route is chosen is $\frac{27}{43}$. Find the probability that I get to school punctually if</p>
<ol class="i-list"><li><div class="list-item-paragraph">I choose the first route,<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">I choose the second route.<span class="inline-marks">[2]</span></div></li></ol>

I have a choice of two routes to get to school. The probability that I am punctual for school is $\frac{43}{50}$ and the probability that I choose the first route is $\frac{3}{5}$ . If I get to school punctually, the probability that the first route is chosen is $\frac{27}{43}$. Find the probability that I get to school punctually if

2013 YJC P2 Q7

<p>The table below shows the number of male and female students studying Chemistry, Physics and Biology at a private school.</p>
<div class="content-block content-block--table"><div class="table-container"><table><colgroup><col style="width: 110px;"><col style="width: 110px;"><col style="width: 110px;"><col style="width: 110px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Chemistry</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Physics</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Biology</p></th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Male</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$200$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$130$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$70$</p></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>Female</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$250$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$300$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$50$</p></td></tr></tbody></table></div></div>
<p>One of the students is chosen at random. Events $C$, $B$ and $M$ are defined as follows:</p>
<p>$C$ : The student chosen is studying Chemistry.</p>
<p>$B$ : The student chosen is studying Biology.</p>
<p>$M$ : The student chosen is a male.</p>
<p>Find</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\mathrm{P}\left( C|M \right)$,<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">$\mathrm{P}\left( M\cup C \right)$,<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">$\mathrm{P}\left( M'\cap B' \right)$.<span class="inline-marks">[1]</span></div></li></ol>
<p>Determine whether $C$ and $M$ are independent.<span class="inline-marks">[2]</span></p>
<p>It is given that $20\%$ of Chemistry students, $30\%$ of Physics students and $5\%$ of Biology students are international students.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">One of the students selected at random is an international student. What is the probability that this student studies Chemistry?<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Three students are chosen at random. Find the probability that there is exactly one international student who studies Physics.<span class="inline-marks">[2]</span></div></li></ol>

The table below shows the number of male and female students studying Chemistry, Physics and Biology at a private school.

One of the students is chosen at random. Events $C$, $B$ and $M$ are defined as follows:

$C$ : The student chosen is studying Chemistry.

$B$ : The student chosen is studying Biology.

Determine whether $C$ and $M$ are independent.

It is given that $20\%$ of Chemistry students, $30\%$ of Physics students and $5\%$ of Biology students are international students.

One of the students selected at random is an international student. What is the probability that this student studies Chemistry?

Three students are chosen at random. Find the probability that there is exactly one international student who studies Physics.

Independent Events

Conditional Probability from Tables

Addition Rule for Overlapping Events

2016 JJC P2 Q6

<p>In this question you should assume that $T$, $W$ and $D$ follow independent normal distributions.</p>
<p>James leaves home to go to work at $T$ minutes past 8 am each day, where $T$ follows the distribution $N\left( 5,\ {{1.2}^{2}} \right)$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Sketch this distribution for the period from 8 am to 8.10 am.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that, on a randomly chosen day, James leaves for work later than 8.06 am.</span></div><p style="text-align: right;">[1]</p></li></ol>
<p>When the weather is fine, James walks to work. The time, $W$ minutes, he takes to walk to work follows the distribution $N\left( 21,\ {{3}^{2}} \right)$. James is supposed to start work at 8.30 am.</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the probability that, on a randomly chosen day when James walks, he is late for work.</span></div><p style="text-align: right;">[2]</p></li></ol>
<p>When the weather is not fine, James drives to work. He still leaves at $T$ minutes past 8 am each day; the time, $D$ minutes, he takes to drive to work follows the distribution $N\left( 19,\ {{6}^{2}} \right)$.</p>
<p>On average, the weather is fine on $70\%$ of mornings.</p>
<ol class="i-list" start="4"><li><div class="list-item-content"><span class="list-item-text">One day, James is late for work. Find the probability that the weather is fine that day.</span></div><p style="text-align: right;">[5]</p></li></ol>

In this question you should assume that $T$, $W$ and $D$ follow independent normal distributions.

James leaves home to go to work at $T$ minutes past 8 am each day, where $T$ follows the distribution $N\left( 5,\ {{1.2}^{2}} \right)$.

Sketch this distribution for the period from 8 am to 8.10 am.

Find the probability that, on a randomly chosen day, James leaves for work later than 8.06 am.

When the weather is fine, James walks to work. The time, $W$ minutes, he takes to walk to work follows the distribution $N\left( 21,\ {{3}^{2}} \right)$. James is supposed to start work at 8.30 am.

Find the probability that, on a randomly chosen day when James walks, he is late for work.

When the weather is not fine, James drives to work. He still leaves at $T$ minutes past 8 am each day; the time, $D$ minutes, he takes to drive to work follows the distribution $N\left( 19,\ {{6}^{2}} \right)$.

On average, the weather is fine on $70\%$ of mornings.

One day, James is late for work. Find the probability that the weather is fine that day.

Normal Distribution: Model and Parameters

RI Normal Distribution Tutorial Q8

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Suppose a fair die is tossed twice. Calculate the probabilities that</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the sum of the scores of the two tosses is at least $8$,</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">the absolute difference between the scores of the two tosses is at least $4$.</span></div><p style="text-align: right;">[1]</p></li></ol></li></ol>
<p>In one round of a game, a player is to draw a ball, without replacement, from a box that contains $3$ red balls and $4$ white balls. If a red ball is drawn, the player will add the scores obtained from tossing a fair die twice. If a white ball is drawn, the player will take the absolute difference of the scores obtained from tossing a die twice.</p>
<p>The game ends if the sum of the scores is at least 8 or the absolute difference of the scores is at least $4$. Else, the player will proceed to the second round of the game where the process of picking a ball from the box and tossing the die twice repeats.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the probability that the game ends at the first round.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Suppose the game ends at the first round. Find the probability that a red ball is drawn.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that there are a total of $3$ rounds of game played and exactly $2$ white balls are selected.</span></div><p style="text-align: right;">[3]</p></li></ol>

Suppose a fair die is tossed twice. Calculate the probabilities that

In one round of a game, a player is to draw a ball, without replacement, from a box that contains $3$ red balls and $4$ white balls. If a red ball is drawn, the player will add the scores obtained from tossing a fair die twice. If a white ball is drawn, the player will take the absolute difference of the scores obtained from tossing a die twice.

The game ends if the sum of the scores is at least 8 or the absolute difference of the scores is at least $4$. Else, the player will proceed to the second round of the game where the process of picking a ball from the box and tossing the die twice repeats.

Find the probability that the game ends at the first round.

Suppose the game ends at the first round. Find the probability that a red ball is drawn.

Find the probability that there are a total of $3$ rounds of game played and exactly $2$ white balls are selected.

Probability Using Possibility Diagrams

Waiting-Time and Stopping-Rule Probabilities

2016 NJC P2 Q10

<p>John plays for his school’s soccer team. There is a probability of $0.15$ that he scores in a game and a probability of $0.3$ that his parents are present at a game. When he scores in a game, there is a probability of $0.2$ that his parents are present.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that the probability that he scores in a game when his parents are present is $0.1$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">State, with justification, whether his parents’ presence at a game will affect his chances of scoring in the game.</span></div><p style="text-align: right;">[1]</p></li></ol>
<p>Games are equally likely to be home or away games. In a home game, there is a probability of $0.24$ that John does not score and his parents are present.</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the least and greatest values of the probability that a game is a home game and his parents are not present at the game.</span></div><p style="text-align: right;">[3]</p></li></ol>

John plays for his school’s soccer team. There is a probability of $0.15$ that he scores in a game and a probability of $0.3$ that his parents are present at a game. When he scores in a game, there is a probability of $0.2$ that his parents are present.

Show that the probability that he scores in a game when his parents are present is $0.1$.

State, with justification, whether his parents’ presence at a game will affect his chances of scoring in the game.

Games are equally likely to be home or away games. In a home game, there is a probability of $0.24$ that John does not score and his parents are present.

Find the least and greatest values of the probability that a game is a home game and his parents are not present at the game.

Probability Bounds and Inequalities

2016 DHS P2 Q6

<p>In a probability experiment, three containers have the following contents.</p>
<p>A jar contains ${2}$ white dice and ${3}$ black dice.</p>
<p>A white box contains ${5}$ red balls and ${3}$ green balls.</p>
<p>A black box contains ${4}$ red balls and ${3}$ green balls.</p>
<p>One die is taken at random from the jar. If the die is white, two balls are taken from the white box, at random and without replacement. If the die is black, two balls are taken from the black box, at random and without replacement.</p>
<p>Events ${W}$ and ${M}$ are defined as follows.</p>
<p>${W: }$ A white die is taken from the jar.</p>
<p>${ M: }$ One red ball and one green ball are obtained.</p>
<p>Show that $\mathrm{P}\left( M\left| W \right. \right)=\frac{15}{28}$.</p><p style="text-align: right;">[2]</p>
<p>Find, giving each of your answers as an exact fraction in its lowest terms,</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">${P(M \cap W)}$,</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}\left( W\left| M \right. \right)$,</span></div><p style="text-align: right;">[5]</p></li><li><div class="list-item-content"><span class="list-item-text">${P(W \cup M)}$.</span></div><p style="text-align: right;">[3]</p></li></ol>
<p>All the dice and balls are now placed in a single container, and four objects are taken at random, each object being replaced before the next one is taken. Find the probability that one object of each colour is obtained.</p>

In a probability experiment, three containers have the following contents.

A jar contains ${2}$ white dice and ${3}$ black dice.

A white box contains ${5}$ red balls and ${3}$ green balls.

A black box contains ${4}$ red balls and ${3}$ green balls.

One die is taken at random from the jar. If the die is white, two balls are taken from the white box, at random and without replacement. If the die is black, two balls are taken from the black box, at random and without replacement.

Events ${W}$ and ${M}$ are defined as follows.

${W: }$ A white die is taken from the jar.

${ M: }$ One red ball and one green ball are obtained.

Show that $\mathrm{P}\left( M\left| W \right. \right)=\frac{15}{28}$.

Find, giving each of your answers as an exact fraction in its lowest terms,

$\mathrm{P}\left( W\left| M \right. \right)$,

All the dice and balls are now placed in a single container, and four objects are taken at random, each object being replaced before the next one is taken. Find the probability that one object of each colour is obtained.

Math Question Bank