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PSLE

PSLE Foundation Mathematics

<p>One round of a navigation game is played by moving a token from the starting point $P$ to one of the finishing points on the right-hand edge of the grid shown.</p>
<ul class="bullet-list"><li data-list-item-id="rule-1" data-list-item-path="rule-1"><div class="list-item-paragraph">The token may only move one grid spacing to the right or one grid spacing upwards at each move.</div></li><li data-list-item-id="rule-2" data-list-item-path="rule-2"><div class="list-item-paragraph">At every move after $P$, the token moves to the right with probability $p$ or upwards with probability $q$, where&nbsp;$p+q=1$. The moves are independent.</div></li><li data-list-item-id="rule-3" data-list-item-path="rule-3"><div class="list-item-paragraph">To win a round, the token must pass through $C$ and finish at $T$.</div></li></ul>
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<ol class="abc-list"><li data-list-item-id="part-a-item" data-list-item-path="part-a-item"><div class="list-item-paragraph">Show that the probability of winning a round is $45p^6q^3$.<span class="inline-marks">[3]</span></div></li></ol>
<p>It is given that&nbsp;$p=\frac34$. Aisha plays one game comprising 16 rounds. Each round is independent of the others. Let&nbsp;$W$ be the number of rounds that Aisha wins in one game.</p>
<ol class="abc-list" start="2"><li data-list-item-id="part-b-item" data-list-item-path="part-b-item"><div class="list-item-paragraph">Find the probability that Aisha wins at least 4 of the 16 rounds.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="part-c-item" data-list-item-path="part-c-item"><div class="list-item-paragraph"><p>Aisha plays 50 such games independently. Estimate the probability that the mean number of rounds she wins per game is at most 1.8. Give your answer to 3 significant figures.</p><span class="inline-marks">[4]</span></div></li></ol>

One round of a navigation game is played by moving a token from the starting point $P$ to one of the finishing points on the right-hand edge of the grid shown.

The token may only move one grid spacing to the right or one grid spacing upwards at each move.

At every move after $P$, the token moves to the right with probability $p$ or upwards with probability $q$, where $p+q=1$. The moves are independent.

To win a round, the token must pass through $C$ and finish at $T$.

Show that the probability of winning a round is $45p^6q^3$.

It is given that $p=\frac34$. Aisha plays one game comprising 16 rounds. Each round is independent of the others. Let $W$ be the number of rounds that Aisha wins in one game.

Find the probability that Aisha wins at least 4 of the 16 rounds.

Central Limit Theorem

Binomial Modelling and Applications

Navigation game through a checkpoint

Sampling

Discrete Random Variables (DRV)

<p>Bluewater Ferries operates a service with $180$ passenger seats. To reduce the number of empty seats, the company may accept more reservations than the number of seats available.</p>
<p>Each reservation costs $\$72$ and is non-refundable. Past records show that a reservation holder does not attend with probability $0.07$. If more than $180$ reservation holders attend, the company pays $\$320$ in compensation to each passenger who is unable to board.</p>
<p>Assume that reservation holders attend independently and that the attendance probability is the same for every reservation. For a particular crossing, the company accepts $190$ reservations. Let&nbsp;$X$ denote the number of reservation holders who attend.</p>
<ol class="abc-list"><li data-list-item-id="q2-a" data-list-item-path="q2-a"><div class="list-item-paragraph">Find the probability that there are insufficient seats for the crossing.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="q2-b" data-list-item-path="q2-b"><div class="list-item-paragraph">Calculate the expected net revenue from the crossing, giving your answer to the nearest dollar.<span class="inline-marks">[3]</span></div></li></ol>
<p>Suppose instead that the company accepts $n$ reservations, where&nbsp;$n&gt;180$, and let&nbsp;$X\sim\mathrm{B}(n,0.93)$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="q2-c" data-list-item-path="q2-c"><div class="list-item-paragraph">Write down an expression, in terms of $n$ and binomial probabilities, for the expected net revenue from one crossing.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="q2-d" data-list-item-path="q2-d"><div class="list-item-paragraph">Hence, determine the number of reservations that the company should accept to maximise the expected net revenue, and state this maximum expected net revenue to the nearest dollar.<span class="inline-marks">[3]</span></div></li></ol>

Bluewater Ferries operates a service with $180$ passenger seats. To reduce the number of empty seats, the company may accept more reservations than the number of seats available.

Each reservation costs $\$72$ and is non-refundable. Past records show that a reservation holder does not attend with probability $0.07$. If more than $180$ reservation holders attend, the company pays $\$320$ in compensation to each passenger who is unable to board.

Assume that reservation holders attend independently and that the attendance probability is the same for every reservation. For a particular crossing, the company accepts $190$ reservations. Let $X$ denote the number of reservation holders who attend.

Suppose instead that the company accepts $n$ reservations, where $n>180$, and let $X\sim\mathrm{B}(n,0.93)$.

Ferry overbooking and expected net revenue

<p>The Horizon Planetarium has&nbsp;$96$ seats for each evening show. To reduce the number of empty seats, it accepts more reservations than the number of seats available. Past records show that a reservation holder attends the show with probability $0.88$. For a particular evening show, $106$ reservations have been accepted. Let&nbsp;$X$ denote the number of reservation holders who attend.</p>
<ol class="abc-list"><li data-list-item-id="q1-a" data-list-item-path="q1-a"><div class="list-item-paragraph">State two assumptions needed for&nbsp;$X$ to be well modelled by a binomial distribution.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="q1-b" data-list-item-path="q1-b"><div class="list-item-paragraph">Assume that the assumptions in part <strong>(a)</strong> hold. Find the probability that there are insufficient seats for the show.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="q1-c" data-list-item-path="q1-c"><div class="list-item-paragraph">Determine the maximum number of reservations that the planetarium may accept so that the probability of having insufficient seats is less than $2\%$.<span class="inline-marks">[3]</span></div></li></ol>

The Horizon Planetarium has $96$ seats for each evening show. To reduce the number of empty seats, it accepts more reservations than the number of seats available. Past records show that a reservation holder attends the show with probability $0.88$. For a particular evening show, $106$ reservations have been accepted. Let $X$ denote the number of reservation holders who attend.

Planetarium reservation limit under a risk constraint

<p>In a school with $1200$ students, an average of $1$ out of $20$ students is sick in a week.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">State, in this context, two assumptions needed for the number of students who are sick in a week to be well modelled by a binomial distribution. Explain why one of these assumptions may not be met.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>For the rest of this question, assume that the number of students who are sick in a week has a binomial distribution.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the expected number of students who are sick in a week.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">In a class of $25$ students, find the probability that at least $2$ of them are sick in a week.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">In a sample of $10$ classes with $25$ students each, find the probability that less than five classes have at least $2$ students who are sick in a week, given that there are at least two such classes.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the least value of $n$ such that in a sample of $n$ randomly selected students, the probability that at least $6$ are sick in a week is more than $0.95$.</span><span class="inline-marks">[3]</span></div></li></ol>

In a school with $1200$ students, an average of $1$ out of $20$ students is sick in a week.

State, in this context, two assumptions needed for the number of students who are sick in a week to be well modelled by a binomial distribution. Explain why one of these assumptions may not be met.

For the rest of this question, assume that the number of students who are sick in a week has a binomial distribution.

Find the expected number of students who are sick in a week.

In a class of $25$ students, find the probability that at least $2$ of them are sick in a week.

In a sample of $10$ classes with $25$ students each, find the probability that less than five classes have at least $2$ students who are sick in a week, given that there are at least two such classes.

Find the least value of $n$ such that in a sample of $n$ randomly selected students, the probability that at least $6$ are sick in a week is more than $0.95$.

Math Question Bank