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PSLE

PSLE Foundation Mathematics

The masses of mangoes in kg have the distribution $\mathrm N(0.3,0.1^2)$.

Find the probability that two randomly selected mangoes weigh at least 0.5 kg in total.

Mark sells fruit boxes, each of which contain 10 randomly selected blueberries and 2 randomly selected mangoes. Let the number of unblemished blueberries in a randomly selected fruit box be $U$. On average, the probability of a blueberry being unblemished is $p$.

State a reason, in the context of the question, why $U$ might not be well-modelled by a binomial distribution.

Assume now that $U$ has the distribution $\mathrm B(10,p)$. The probability that a randomly selected fruit box contains at least 8 unblemished blueberries is 0.9.

Each fruit box is classified as grade A, B or C according to the following criteria:

Show that the probability that a randomly chosen fruit box is classified as grade C is 0.316, correct to 3 significant figures.

Hence find the probability that a randomly selected grade C fruit box contains at least 8 unblemished blueberries.

The selling prices of a grade A, grade B and grade C fruit box are $\$12$, $\$10$ and $\$6$ respectively. Find the probability that the total selling price of 5 randomly selected fruit boxes is less than $\$37$.

Mark decides to amend the criteria for grade A boxes as follows:

**Each** mango in the box weighs at least 0.3 kg **and** the box contains at least 9 unblemished blueberries.

State how the percentage of fruit boxes classified as grade A will change, and explain your answer.

Binomial Models and Assumptions

Symmetry and Properties of the Normal Distribution

Applying Probability Laws

Linear Combinations of Independent Normal Variables

Determining an Unknown Binomial Success Probability

2026 EJC P2 Q11

Probability

Discrete Random Variables (DRV)

Continuous Random Variables

<p>The random variable $X$ has the binomial distribution $\mathrm{B}(n,p)$, where&nbsp;$n\geq2$ and $0&lt;p&lt;1$. The mean of $X$ is three times its standard deviation.</p>
<ol class="abc-list"><li data-list-item-id="bimodal-mean-sd-q2-a" data-list-item-path="bimodal-mean-sd-q2-a"><div class="list-item-paragraph">Show that&nbsp;$p=\frac{9}{n+9}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="bimodal-mean-sd-q2-b" data-list-item-path="bimodal-mean-sd-q2-b"><div class="list-item-paragraph">Given that&nbsp;$X$ is bimodal and $n&lt;40$, find all possible values of $n$ and the corresponding modes of $X$.<span class="inline-marks">[4]</span></div></li></ol>

The random variable $X$ has the binomial distribution $\mathrm{B}(n,p)$, where $n\geq2$ and $0<p<1$. The mean of $X$ is three times its standard deviation.

Bimodal Binomial Distributions

Mean and Variance of a Binomial Distribution

Two modes under a mean-standard-deviation condition

<p>The random variable $X$ has the binomial distribution $\mathrm{B}(n,p)$, where&nbsp;$n$ is a positive integer and $0&lt;p&lt;1$. Denote $\mathrm{P}(X=r)$ by $q_r$.</p>
<ol class="abc-list"><li data-list-item-id="bimodal-mean-q1-a" data-list-item-path="bimodal-mean-q1-a"><div class="list-item-paragraph">Show that&nbsp;$\frac{q_r}{q_{r-1}}=\frac{(n-r+1)p}{r(1-p)}$, for&nbsp;$r=1,2,\ldots,n$. Hence show that&nbsp;$X$ is bimodal if and only if&nbsp;$(n+1)p$ is an integer, and state its two modes in terms of $n$ and $p$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="bimodal-mean-q1-b" data-list-item-path="bimodal-mean-q1-b"><div class="list-item-paragraph">It is known that the modes of $X$ are $11$ and $12$, and that&nbsp;$\mathrm{E}(X)=11.52$. Determine $n$ and $p$.<span class="inline-marks">[3]</span></div></li></ol>

The random variable $X$ has the binomial distribution $\mathrm{B}(n,p)$, where $n$ is a positive integer and $0<p<1$. Denote $\mathrm{P}(X=r)$ by $q_r$.

Binomial Probabilities Using the Probability Mass Function

Two modes from a probability ratio and mean

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">The random variable $Y$ has a binomial distribution with mean $1.6$ and $\mathrm{P}(Y=0)=0.1296$. Find $\mathrm{P}(Y&gt;2)$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">A car park has a large number of parking lots. $80$ parking lots are observed and the number of occupied parking lots is denoted by $X$. State, in context, two assumptions needed for&nbsp;$X$ to be well modelled by a binomial distribution.</div></li></ol>
<p>The probability of a parking lot being occupied is $0.95$. Find the probability that out of the $80$ chosen parking lots, at least $90$% are occupied.</p>

The random variable $Y$ has a binomial distribution with mean $1.6$ and $\mathrm{P}(Y=0)=0.1296$. Find $\mathrm{P}(Y>2)$.

A car park has a large number of parking lots. $80$ parking lots are observed and the number of occupied parking lots is denoted by $X$. State, in context, two assumptions needed for $X$ to be well modelled by a binomial distribution.

The probability of a parking lot being occupied is $0.95$. Find the probability that out of the $80$ chosen parking lots, at least $90$% are occupied.

Binomial Probabilities Using Technology

2013 MJC P2 Q8 [Modified]

<p>Given that $X\sim\mathrm{B}(8,p)$ and $\mathrm{P}(X&lt;3)=0.70573$. Find the value of $p$.</p>

Given that $X\sim\mathrm{B}(8,p)$ and $\mathrm{P}(X<3)=0.70573$. Find the value of $p$.

<p>Given that $X\sim\mathrm{B}(10,p)$ and $\mathrm{P}(X=6)=0.17843$. Find the possible values of $p$.</p>

Given that $X\sim\mathrm{B}(10,p)$ and $\mathrm{P}(X=6)=0.17843$. Find the possible values of $p$.

<p>Given that $X\sim\mathrm{B}(5,p)$ and $\mathrm{P}(X=3)=0.12387$. Find the possible values of $p$.</p>

Given that $X\sim\mathrm{B}(5,p)$ and $\mathrm{P}(X=3)=0.12387$. Find the possible values of $p$.

<p>Given that $X\sim\mathrm{B}(25,p)$ and $\mathrm{Var}(X)=4.49$. Find the possible values of $p$.</p>

Given that $X\sim\mathrm{B}(25,p)$ and $\mathrm{Var}(X)=4.49$. Find the possible values of $p$.

<p>Given that $X\sim\mathrm{B}(25,p)$ and $\mathrm{E}(X)=13.5$. Find $p$.</p>

Given that $X\sim\mathrm{B}(25,p)$ and $\mathrm{E}(X)=13.5$. Find $p$.

<p>Given that $X\sim\mathrm{B}(10,p)$ and $\mathrm{P}(X=6)=0.17843$. Find the possible values of $p$.</p>

Worked Example 3

<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.

Conditional Probability

Probabilities of Successive Events

Repeated Binomial Experiments

Waiting-Time and Stopping-Rule Probabilities

2021 NJC P2 Q8

<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>For a positive integer $n$, it is given that&nbsp;$\mathrm{P}\left( A=n \right)=0.009542$ and $\mathrm{P}\left( A=n+1 \right)=0.004090$, where&nbsp;$A\sim\mathrm{B}\left( 2n+1,p \right)$. Show that&nbsp;$p$ satisfies an equation of the form $\frac{p}{1-p}=k$, where&nbsp;$k$ is a constant to be determined. Hence find the value of $n$ and the variance of $A$.<span class="inline-marks">[4]</span></p>

For a positive integer $n$, it is given that $\mathrm{P}\left( A=n \right)=0.009542$ and $\mathrm{P}\left( A=n+1 \right)=0.004090$, where $A\sim\mathrm{B}\left( 2n+1,p \right)$. Show that $p$ satisfies an equation of the form $\frac{p}{1-p}=k$, where $k$ is a constant to be determined. Hence find the value of $n$ and the variance of $A$.

2024 NJC P2 Q6

<p>The random variable $X$ has distribution $\mathrm{B}\left( n,p \right)$, where&nbsp;$n\ge 2$. It is known that the mean of $X$ and the standard deviation of $X$ are equal.</p>
<ol class="abc-list"><li data-list-item-id="item-1760845825629-mf2jn5n3h" data-list-item-path="item-1760845825629-mf2jn5n3h"><div class="list-item-paragraph">Find an expression for&nbsp;$p$ in terms of $n$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760845827122-fg268dk4z" data-list-item-path="item-1760845827122-fg268dk4z"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{P}\left( X=1 \right)}{\mathrm{P}\left( X=2 \right)}=\frac{2n}{n-1}$. Hence find the the mode(s) of $X$.<span class="inline-marks">[4]</span></div></li></ol>

The random variable $X$ has distribution $\mathrm{B}\left( n,p \right)$, where $n\ge 2$. It is known that the mean of $X$ and the standard deviation of $X$ are equal.

Mode of a Binomial Distribution

2024 EJC P2 Q6

<p>The random variable $X$ follows a binomial distribution $\mathrm{B}\left( 15,p \right)$.</p>
<p>Given that $X$ has two modes, $X=5$ and $X=6$, find the exact value of $p$.</p>

The random variable $X$ follows a binomial distribution $\mathrm{B}\left( 15,p \right)$.

Given that $X$ has two modes, $X=5$ and $X=6$, find the exact value of $p$.

Prelims Revision Set: Bimodal Distribution Q1

<p>The results of an experiment show that the discrete random variable $X$ has a binomial distribution with mean $1.2$ and variance $1.18$.</p>
<p>Find the smallest integer $r$ such that $\mathrm{P}(X &lt; r)&gt;0.85$.</p>

The results of an experiment show that the discrete random variable $X$ has a binomial distribution with mean $1.2$ and variance $1.18$.

Find the smallest integer $r$ such that $\mathrm{P}(X < r)>0.85$.

<p>Always Cool Company produces reusable ice pack gels. It is known that, on average, $p\%$ of the reusable ice pack gels leak. Each day the quality control department takes a random sample of $120$ reusable ice pack gels and checks them for leakage. The number of reusable ice pack gels that leak in a sample is denoted by the random variable $X$.</p>
<ol class="i-list"><li data-list-item-id="item-1760688228763-uzpaivtwb" data-list-item-path="item-1760688228763-uzpaivtwb"><div class="list-item-paragraph">State, in the context of the question, two assumptions needed for&nbsp;$X$ to be well modelled by a binomial distribution.<span class="inline-marks">[2]</span></div></li></ol>
<p>Assume now that&nbsp;$X$ can be modelled by a binomial distribution,</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1760688247723-lbq7o4hrc" data-list-item-path="item-1760688247723-lbq7o4hrc"><div class="list-item-paragraph">It is given that the most likely number of reusable ice pack gels that leak in a sample is $3$. Find the range of values of $p$.<span class="inline-marks">[3]</span></div></li></ol>
<p><strong>For the rest of the question, the value of $p$ is taken to be $3$.</strong></p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760688271407-x2zz1jj7g" data-list-item-path="item-1760688271407-x2zz1jj7g"><div class="list-item-paragraph">Find the probability that the quality control department finds at most $4$ reusable ice pack gels that leaks in a sample.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760688273411-s4xldhdti" data-list-item-path="item-1760688273411-s4xldhdti"><div class="list-item-paragraph">If the quality control department found at least $5$ reusable ice pack gels that leak in a sample, find the probability that there are fewer than $9$ ice pack gels that leak in a sample.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760688281300-ih9acns3w" data-list-item-path="item-1760688281300-ih9acns3w"><div class="list-item-paragraph">The number of reusable ice pack gels that leak in a sample is independent of that of other samples. Find the probability that, in $10$ randomly chosen samples, the eighth sample is the third sample with at most $4$ reusable ice pack gels that leak.<span class="inline-marks">[3]</span></div></li></ol>

Always Cool Company produces reusable ice pack gels. It is known that, on average, $p\%$ of the reusable ice pack gels leak. Each day the quality control department takes a random sample of $120$ reusable ice pack gels and checks them for leakage. The number of reusable ice pack gels that leak in a sample is denoted by the random variable $X$.

Assume now that $X$ can be modelled by a binomial distribution,

**For the rest of the question, the value of $p$ is taken to be $3$.**

2020 MI P2 Q9

<p>A dart board is designed with an inner radius of $r$ cm and an outer radius of $15$ cm, as shown in the diagram 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/converted/1760448997302-lpx6r8fk7mi.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA0NDg5OTczMDItbHB4NnI4Zms3bWkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU2MjIyNCwiZXhwIjoxNzkzNTYyNTI0fQ.nWknZ-BJ2tzofDi6kp3olSkqJVZVSc13xAZUo8U_TD0" style="max-width: min(370px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>A dart throw is considered a ‘hit’ if it lands within the inner circle. It is assumed that any dart throw lands randomly on the dart board.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that the probability of a ‘hit’ is $\frac{{{r}^{2}}}{225}$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Wei plays a round of $8$ dart throws.</div><ol class="i-list"><li><div class="list-item-paragraph">The probability that Wei obtains more than six ‘hits’ is at most $0.08$. Write down an inequality satisfied by $r$. Hence find the range of possible values of $r$.<span class="inline-marks">[2]</span></div><div class="list-item-paragraph">It is now given that&nbsp;$r=6$</div></li><li><div class="list-item-paragraph">Find the probability that Wei’s fifth throw is the second time that he obtains a ‘hit’.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the least number of throws that Wei needs to attempt such that the probability of obtaining at least $1$ ‘hit’ from the attempts is at least $0.7$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Chan also plays a round of $8$ dart throws. The number of ‘hits’ Chan obtained is more than the expected number of ‘hits’ Wei obtained. Find the probability that the number of ‘hits’ Chan obtained is fewer than twice the expected number of ‘hits’ Wei obtained.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

A dart board is designed with an inner radius of $r$ cm and an outer radius of $15$ cm, as shown in the diagram below.

A dart throw is considered a ‘hit’ if it lands within the inner circle. It is assumed that any dart throw lands randomly on the dart board.

Minimum Number of Binomial Trials

2023 YIJC P2 Q9

<ol class="abc-list"><li><div class="list-item-paragraph">The random variable $X$ has the binomial distribution $\mathrm{B}\left( 10,p \right)$ such that&nbsp;$\mathrm{P}(X=2)=0.0325$.<span class="inline-marks">[3]</span></div><div class="list-item-paragraph">Write down an equation in terms of $p$ and find the value of p given that&nbsp;$p&lt;0.4$.</div></li></ol>

2013 IJC P2 Q8

<p>In a school with a large number of students, $p\%$, where&nbsp;$p&gt;40$, of its students study H2 Biology. In a random sample of $40$ students, the probability of no less than nine but no more than twenty students studying H2 Biology is $0.25$. Find the value of $p$.<span class="inline-marks">[3]</span></p>
<p>It is given that&nbsp;$p=60$. A group of $12$ randomly chosen students are to attend an interview session at a meeting room, with only one interviewee being interviewed at a time. Find the probability that the third and tenth interviewee are the first and fourth students respectively among them studying H2 Biology.<span class="inline-marks">[3]</span></p>

In a school with a large number of students, $p\%$, where $p>40$, of its students study H2 Biology. In a random sample of $40$ students, the probability of no less than nine but no more than twenty students studying H2 Biology is $0.25$. Find the value of $p$.

It is given that $p=60$. A group of $12$ randomly chosen students are to attend an interview session at a meeting room, with only one interviewee being interviewed at a time. Find the probability that the third and tenth interviewee are the first and fourth students respectively among them studying H2 Biology.

Math Question Bank