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<p>Airlines routinely overbook flights based on the expectation that some fraction of booked passengers will not show up for each flight.For a particular flight that has a maximum capacity of $300$ seats, the airline has sold $312$ tickets.Assume that a booked passenger will not show for the flight with probability $5\%$ and that the number of passengers who show up for each flight follows a binomial distribution.</p>
<ol class="i-list"><li data-list-item-id="item-1760702896205-j2u1p6b9e" data-list-item-path="item-1760702896205-j2u1p6b9e"><div class="list-item-paragraph">What is the probability that there will not be enough seats for the flight?</div></li><li data-list-item-id="item-1760702896963-chmaz9d5l" data-list-item-path="item-1760702896963-chmaz9d5l"><div class="list-item-paragraph">Find the probability that the flight has empty seat(s).</div></li><li data-list-item-id="item-1760702898022-vyceql7w9" data-list-item-path="item-1760702898022-vyceql7w9"><div class="list-item-paragraph">Find the number of tickets that can be sold to ensure that the probability of overbooking is less than $3\%$.</div></li></ol>

Airlines routinely overbook flights based on the expectation that some fraction of booked passengers will not show up for each flight.For a particular flight that has a maximum capacity of $300$ seats, the airline has sold $312$ tickets.Assume that a booked passenger will not show for the flight with probability $5\%$ and that the number of passengers who show up for each flight follows a binomial distribution.

Binomial Probabilities Using Technology

Maximum Number of Binomial Trials

Prelims Revision Set: Problems involving Overbooking Q1

Discrete Random Variables (DRV)

<p>The random variable $X$ is normally distributed with mean $\mu $ and variance ${{\sigma }^{2}}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760683461527-ea96b8ekp" data-list-item-path="item-1760683461527-ea96b8ekp"><div class="list-item-paragraph">Find $\mathrm{P}\left( X&gt;\mu +0.05\sigma \right)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760683462143-lzfp5v0xe" data-list-item-path="item-1760683462143-lzfp5v0xe"><div class="list-item-paragraph">The mean of a random sample of $n$ independent observations of $X$ is denoted by $\bar{X}$.</div><div class="list-item-paragraph">Find the least value of $n$ such that the probability that&nbsp;$\bar{X}$ differs from $\mu $ by at least $0.05\sigma $ is less than $0.4$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760683463553-6e6h0ye4h" data-list-item-path="item-1760683463553-6e6h0ye4h"><div class="list-item-paragraph">In another random sample of $m$ independent observations of $X$, the probability that at least $4$ of these observations exceeds $\mu +0.05\sigma $ each is at most $\mathrm{0}\mathrm{.95}$. Find the greatest value of $m$.<span class="inline-marks">[3]</span></div></li></ol>

The random variable $X$ is normally distributed with mean $\mu $ and variance ${{\sigma }^{2}}$.

Solving Normal Probability Inequalities

Determining Sample Size from Probability Conditions

Normal Models for Sample Means and Averages

Standardising Normal Random Variables

2023 NJC P2 Q6

Sampling

Continuous Random Variables

<p>On the average, $30\%$ of the students in Calculus Madness Institute (CMI) could do the Differential Equation question in Block Test Two. The principal randomly select a class of $30$ students to analyse the results.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, in the context of this question, two assumptions needed to model the results by a binomial distribution.</div></li><li><div class="list-item-paragraph">Find the probability that at least $6$ students in the class could do that question.</div></li><li><div class="list-item-paragraph">Find the probability that only $2$ students among the first $8$ selected students in the class could do the question given that at least $6$ students could do that question.</div></li><li><div class="list-item-paragraph">The probability of no more than $5$ students could do the question in a randomly selected class exceeds $0.9$. Find the largest possible number of students in that class.</div></li></ol>
<p>$50$ classes with $30$ students each are selected randomly. Find the probability that an average of at least $10$ students per class could do the question.</p>

On the average, $30\%$ of the students in Calculus Madness Institute (CMI) could do the Differential Equation question in Block Test Two. The principal randomly select a class of $30$ students to analyse the results.

$50$ classes with $30$ students each are selected randomly. Find the probability that an average of at least $10$ students per class could do the question.

Conditional Probability

Central Limit Theorem

Binomial Models and Assumptions

Sample Mean and Its Sampling Distribution

2012 SAJC P2 Q13

Probability

<p>A biscuit factory produces biscuits which are packed into its signature gift boxes. On average, $3\%$ of these gift boxes packed are substandard.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the probability that a randomly chosen batch of $30$ gift boxes will contain at least $2$ gift boxes which are substandard is $0.227$, correct to $3$ significant figures.</div><p style="text-align: right;">[2]</p></li></ol>
<p>The operations unit of the factory decided to pack $30$ gift boxes into one carton before delivery to all its retail outlets.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the probability that out of $10$ randomly chosen cartons, there are more than $4$ cartons with at least $2$ substandard gift boxes.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the probability that out of $10$ randomly chosen cartons, there are at least $2$ substandard gift boxes.</div><p style="text-align: right;">[2]</p></li></ol>
<p>Every week, the owner of each retail outlet will place order with the factory.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Find the maximum number of cartons to be ordered so that the chance of having no substandard gift box is more than $5\%$.</div><p style="text-align: right;">[4]</p></li></ol>

A biscuit factory produces biscuits which are packed into its signature gift boxes. On average, $3\%$ of these gift boxes packed are substandard.

The operations unit of the factory decided to pack $30$ gift boxes into one carton before delivery to all its retail outlets.

Every week, the owner of each retail outlet will place order with the factory.

Repeated Binomial Experiments

ASRJC Binomial Distribution Tutorial Q12

<p>Based on past statistical data of high speed train journey, there is a $8\%$ chance that a passenger with reservation made will not show up. In order to maximise revenue, a high speed train company accepts more reservations than the passenger capacity of its trains.</p>
<ol class="i-list"><li data-list-item-id="item-1760260797041-fl6nn7gbg" data-list-item-path="item-1760260797041-fl6nn7gbg"><div class="list-item-paragraph">State $2$ assumptions needed such that the number of passengers who do not show up for a high speed train journey may be well modelled by a Binomial distribution.<span class="inline-marks">[2]</span></div></li></ol>
<p>A train company operates a high speed train from Burong Lake to Huahin Cove which has a capacity of $280$ passengers.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1760260811831-77gnmqp97" data-list-item-path="item-1760260811831-77gnmqp97"><div class="list-item-paragraph">Find the probability that when&nbsp;$300$ reservations are accepted, the high speed train journey is overbooked, i.e. there is not enough seats available for the passengers who show up.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760260813009-pm44nye4r" data-list-item-path="item-1760260813009-pm44nye4r"><div class="list-item-paragraph">Find the maximum number of reservations that should be accepted in order to ensure that the probability of overbooking is less than $1\%$.<span class="inline-marks">[3]</span></div></li></ol>
<p>The high speed train operates once daily throughout the year and $300$ reservations are accepted for each journey.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1760260842185-3ymwexvh4" data-list-item-path="item-1760260842185-3ymwexvh4"><div class="list-item-paragraph">Find the probability that no high speed train journey is overbooked in a week.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760260842875-ed4rv7k7m" data-list-item-path="item-1760260842875-ed4rv7k7m"><div class="list-item-paragraph">Two such journeys from Burong Lake to Huahin Cove with $300$ reservations each are examined. If the total number of passengers who turned up for these two journeys is at most $550$ passengers, find the probability that one of the journeys is overbooked by at most $2$ passengers.<span class="inline-marks">[3]</span></div></li></ol>

Based on past statistical data of high speed train journey, there is a $8\%$ chance that a passenger with reservation made will not show up. In order to maximise revenue, a high speed train company accepts more reservations than the passenger capacity of its trains.

A train company operates a high speed train from Burong Lake to Huahin Cove which has a capacity of $280$ passengers.

The high speed train operates once daily throughout the year and $300$ reservations are accepted for each journey.

Probabilities Involving Multiple Binomial Variables

2018 SRJC P2 Q12

<p>On average, $40\%$ of Christmas tree light bulbs manufactured by a company are red and the rest are blue. The lights are sold in boxes of $20$. Assume that the number of red light bulbs in a box has a binomial distribution.</p>
<ol class="i-list"><li data-list-item-id="item-1759565645014-i9cb194l6" data-list-item-path="item-1759565645014-i9cb194l6"><div class="list-item-paragraph">Find the probability that a box of $20$ light bulbs contains fewer red light bulbs than blue light bulbs.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1759565645517-jmh945g6d" data-list-item-path="item-1759565645517-jmh945g6d"><div class="list-item-paragraph">In $n$ randomly chosen boxes, the probability that there will be at least $2$ boxes with fewer red light bulbs than blue is at most $0.999$. Find the greatest value of $n$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759565646799-y2ampdchn" data-list-item-path="item-1759565646799-y2ampdchn"><div class="list-item-paragraph">A random sample of $50$ boxes is chosen. Using an approximate distribution, find the probability that the mean number of red light bulbs in $1$ box will not exceed $7.5$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759565647422-zwwls5vd4" data-list-item-path="item-1759565647422-zwwls5vd4"><div class="list-item-paragraph">A customer selects boxes of light bulbs at random from a large consignment until she finds a box with fewer red lights than blue. Give a reason why a binomial distribution is not an appropriate model for the number of boxes selected.<span class="inline-marks">[1]</span></div></li></ol>

On average, $40\%$ of Christmas tree light bulbs manufactured by a company are red and the rest are blue. The lights are sold in boxes of $20$. Assume that the number of red light bulbs in a box has a binomial distribution.

Find the probability that a box of $20$ light bulbs contains fewer red light bulbs than blue light bulbs.

In $n$ randomly chosen boxes, the probability that there will be at least $2$ boxes with fewer red light bulbs than blue is at most $0.999$. Find the greatest value of $n$.

A random sample of $50$ boxes is chosen. Using an approximate distribution, find the probability that the mean number of red light bulbs in $1$ box will not exceed $7.5$.

A customer selects boxes of light bulbs at random from a large consignment until she finds a box with fewer red lights than blue. Give a reason why a binomial distribution is not an appropriate model for the number of boxes selected.

2018 ACJC P2 Q6

<p>In an opinion poll before an election, a sample of $30$ voters is obtained. The number of voters in the sample who support the Alliance Party is denoted by $A$. State in context, two conditions that must be assumed for $A$ to be modelled by a Binomial Distribution.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">It is given that $5\%$ of the voters in the election support the Alliance Party.</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">What is the probability that fewer than $6$ of the $30$ voters support the Alliance Party?</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">What will be the most likely number of voters who support the Alliance Party in a sample of $40$ voters?</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">It is given that, in a sample of $50$ voters, the probability that at least $m$ voters who support the Alliance Party is less than $0.4$. Find the least possible value of $m$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">If a sample of $n$ number of voters is obtained and the probability of getting at least a supporter is not more than $0.75$. Find the maximum value of $n$.</span></div><p style="text-align: right;">[3]</p></li></ol></li><li><div class="list-item-content"><span class="list-item-text">In the next election, another sample of $30$ voters is obtained and the percentage of the voters is $p$ where $p>50$. It is given that the probability of getting $15$to $20$supporters is $0.3$. Find the value of $p$.</span></div><p style="text-align: right;">[4]</p></li></ol>

In an opinion poll before an election, a sample of $30$ voters is obtained. The number of voters in the sample who support the Alliance Party is denoted by $A$. State in context, two conditions that must be assumed for $A$ to be modelled by a Binomial Distribution.

It is given that $5\%$ of the voters in the election support the Alliance Party.

In the next election, another sample of $30$ voters is obtained and the percentage of the voters is $p$ where $p>50$. It is given that the probability of getting $15$to $20$supporters is $0.3$. Find the value of $p$.

Math Question Bank