Questions Further Paper 4 (133 questions)

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CAIE Further Paper 4 2023 November Q5
16 marks Standard +0.8
5 A company is deciding which of two machines, \(X\) and \(Y\), can make a certain type of electrical component more quickly. The times taken, in minutes, to make one component of this type are recorded for a random sample of 8 components made by machine \(X\) and a random sample of 9 components made by machine \(Y\). These times are as follows.
Machine \(X\)4.04.64.74.85.05.25.65.8
Machine \(Y\)4.54.95.15.35.45.75.96.36.4
The manager claims that on average the time taken by machine \(X\) to make one component is less than that taken by machine \(Y\).
  1. Carry out a Wilcoxon rank-sum test at the \(5 \%\) significance level to test whether the manager's claim is supported by the data.
  2. Assuming that the times taken to produce the components by the two machines are normally distributed with equal variances, carry out a \(t\)-test at the \(5 \%\) significance level to test whether the manager's claim is supported by the data.
    \section*{Question 5(c) is printed on the next page.}
  3. In general, would you expect the conclusions from the tests in parts (a) and (b) to be the same? Give a reason for your answer.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 4 2023 November Q3
8 marks Standard +0.3
3 Scientists are studying the effects of exercise on LDL blood cholesterol levels. Over a three-month period, a large group of people exercised for 20 minutes each day. For a randomly chosen sample of 10 of these people, the LDL blood cholesterol levels were measured at the beginning and the end of the three-month period. The results, measured in suitable units, are as follows.
\cline { 2 - 12 } \multicolumn{1}{c|}{}Person\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)
\multirow{2}{*}{
Cholesterol
level
}		Beginning72841209010213564758088
\cline { 2 - 12 }End64761059210511567757584
  1. Test, at the \(2.5 \%\) significance level, whether there is evidence that the population mean LDL blood cholesterol level has reduced by more than 2 units after the three-month period.
  2. State any assumption that you have made in part (a). \includegraphics[max width=\textwidth, alt={}, center]{b6635fbc-3c9d-4f93-b51a-b1cbd71ddbb1-06_399_1383_269_324} As shown in the diagram, the continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} m x & 0 \leqslant x \leqslant 2 \\ \frac { k } { x ^ { 2 } } + c & 2 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(m , k\) and \(c\) are constants.
CAIE Further Paper 4 2024 November Q1
6 marks Standard +0.3
1 Ellie is investigating the heights of two types of beech tree, \(A\) and \(B\), in a certain region. She has chosen a random sample of 60 beech trees of type \(A\) in the region, recorded their heights, \(x \mathrm {~m}\), and calculated unbiased estimates for the population mean and population variance as 35.6 m and \(4.95 \mathrm {~m} ^ { 2 }\) respectively. Ellie also chooses a random sample of 50 beech trees of type \(B\) in the region and records their heights, \(y \mathrm {~m}\). Her results are summarised as follows. $$\sum y = 1654 \quad \sum y ^ { 2 } = 54850$$ Find a \(95 \%\) confidence interval for the difference between the population mean heights of type \(A\) and type \(B\) beech trees in the region.
CAIE Further Paper 4 2024 November Q2
9 marks Standard +0.3
2 A school with a large number of students is updating its logo. Each student has designed a new logo and two teachers have each awarded a mark out of 50 for each logo. The marks awarded to a random sample of 12 students are shown in the following table.
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)\(K\)\(L\)
Teacher 1363840362234454448352830
Teacher 2384232413241425036444241
One of the students claims that Teacher 2 is awarding higher marks than Teacher 1.
  1. Carry out a Wilcoxon matched-pairs signed-rank test, at the \(5 \%\) significance level, to test whether the data supports the claim. \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-04_2720_38_109_2010} \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-05_2717_29_105_22} It was later discovered that Teacher 1 had entered her mark for student \(C\) incorrectly. Her intended mark was 24 not 40 . This was corrected.
  2. Determine whether this correction affects the conclusion of the test carried out in part (a).
CAIE Further Paper 4 2024 November Q3
8 marks Standard +0.3
3 A statistician believes that the number of telephone calls received by an advice centre in a 10 -minute interval can be modelled by the Poisson distribution \(\mathrm { Po } ( 1.9 )\). The number of calls received in a randomly chosen 10-minute interval was recorded on each of 100 days. The results are summarised in the table, together with some of the expected frequencies corresponding to the distribution \(\operatorname { Po } ( 1.9 )\).
Number of calls0123456 or more
Observed frequency101835211141
Expected frequency14.95728.41826.9971.322
  1. Complete the table.
  2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to determine whether the statistician's belief is reasonable. \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-07_2726_35_97_20}
CAIE Further Paper 4 2024 November Q4
10 marks Standard +0.3
4 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} k x ^ { 3 } & 0 \leqslant x < 1 , \\ k ( 5 - x ) & 1 \leqslant x \leqslant 5 , \\ 0 & \text { otherwise } , \end{cases}$$ where \(k\) is a constant.
  1. Sketch the graph of f.
  2. Show that \(k = \frac { 4 } { 33 }\). \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-09_2725_35_99_20}
  3. Find the cumulative distribution function of \(X\).
  4. Find the median value of \(X\).
CAIE Further Paper 4 2024 November Q5
9 marks Standard +0.8
5 Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 3 }\) and the third coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 5 }\). The random variable \(X\) is the number of heads that Nikita obtains when he throws all three coins at the same time.
  1. Find the probability generating function of \(X\).
    Rajesh has two fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6. The random variable \(Y\) is the number of 4 s that Rajesh obtains when he throws the two dice. The random variable \(Z\) is the sum of the number of heads obtained by Nikita and the number of 4 s obtained by Rajesh.
  2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
    ΝΙΟθW SΙΗΙ ΝΙ ΞιΙΥΜ ιΟΝ Ο0\includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_446_37_674_2013}\includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_444_37_1245_2013}\includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_441_33_1816_2013}\includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_443_33_2387_2013}
    \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-11_2726_35_97_20}
  3. Use your answer to part (b) to find \(\mathrm { E } ( Z )\).
CAIE Further Paper 4 2024 November Q6
8 marks Standard +0.8
6 Ansal is investigating the wingspans of Monarch butterflies in two different regions, \(X\) and \(Y\). He takes a random sample of 8 Monarch butterflies from region \(X\) and records their wingspans, \(x \mathrm {~cm}\). His results are as follows. $$\begin{array} { l l l l l l l l } 8.2 & 7.0 & 7.3 & 8.8 & 7.8 & 8.5 & 9.2 & 7.4 \end{array}$$ Ansal also takes a random sample of 9 Monarch butterflies from region \(Y\) and records their wingspans, \(y \mathrm {~cm}\). His results are summarised as follows. $$\sum y = 71.10 \quad \sum y ^ { 2 } = 567.13$$ Ansal suspects that the mean wingspan of Monarch butterflies from region \(X\) is greater than the mean wingspan of Monarch butterflies from region \(Y\). It is known that the wingspans of Monarch butterflies in regions \(X\) and \(Y\) are normally distributed with equal population variances. Test, at the 10\% significance level, whether Ansal's suspicion is supported by the data. \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-12_2717_35_109_2012} \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-13_2726_35_97_20}
If you use the following page to complete the answer to any question, the question number must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-14_2715_33_109_2012}
CAIE Further Paper 4 2024 November Q1
4 marks Standard +0.3
1 A scientist is investigating the lengths of the leaves of a certain type of plant. The scientist assumes that the lengths of the leaves of this type of plant are normally distributed. He measures the lengths, \(x \mathrm {~cm}\), of the leaves of a random sample of 8 plants of this type. His results are as follows. \(\begin{array} { l l l l l l l l } 3.5 & 4.2 & 3.8 & 5.2 & 2.9 & 3.7 & 4.1 & 3.2 \end{array}\) Find a \(90 \%\) confidence interval for the population mean length of leaves of this type of plant.
CAIE Further Paper 4 2024 November Q2
8 marks Challenging +1.2
2 The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$\mathrm { G } _ { X } ( t ) = \frac { 1 } { 5 } + p t + q t ^ { 2 }$$ where \(p\) and \(q\) are constants.
  1. Given that \(\mathrm { E } ( X ) = 1.1\), find the numerical value of \(\operatorname { Var } ( X )\). \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-04_2714_38_109_2010} The random variable \(Y\) has probability generating function \(\mathrm { G } _ { Y } ( t )\) given by $$\mathrm { G } _ { Y } ( t ) = \frac { 2 } { 3 } t \left( 1 + \frac { 1 } { 2 } t ^ { 2 } \right)$$ The random variable \(Z\) is the sum of independent observations of \(X\) and \(Y\).
  2. Find the probability generating function of \(Z\).
  3. Find \(\mathrm { P } ( Z > 2 )\).
  4. State the most probable value of \(Z\).
CAIE Further Paper 4 2024 November Q3
10 marks Standard +0.3
3 Rosie sows 5 seeds in each of 150 plant pots. The number of seeds that germinate is recorded for each pot. The results are summarised in the following table.
Number of seeds that germinate012345
Number of pots12404335164
Rosie suggests that the number of seeds that germinate follows the binomial distribution \(\mathrm { B } ( 5 , p )\).
  1. Use Rosie's results to show that \(p = 0.42\).
  2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to test whether the distribution \(\mathrm { B } ( 5,0.42 )\) is a good fit for the data. \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-06_2720_38_109_2010} \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-07_2726_35_97_20}
CAIE Further Paper 4 2024 November Q4
10 marks Standard +0.8
4 The random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 21 } ( x - 1 ) ^ { 2 } & 2 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$
  1. Find the cumulative distribution function of \(X\).
    The random variable \(Y\) is defined by \(Y = ( X - 1 ) ^ { 4 }\).
  2. Find the probability density function of \(Y\). \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-09_2725_35_99_20}
  3. Find the median value of \(Y\).
  4. Find \(\mathrm { E } ( Y )\).
CAIE Further Paper 4 2024 November Q5
9 marks Challenging +1.2
5 Dev owns a small company which produces bottles of juice. He uses two machines, \(X\) and \(Y\), to fill empty bottles with juice. Dev is investigating the volumes of juice in the bottles. He chooses a random sample of 35 bottles filled by machine \(X\) and a random sample of 60 bottles filled by machine \(Y\). The volumes of juice, \(x\) and \(y\) respectively, measured in suitable units, are summarised by $$\sum x = 30.8 , \quad \sum x ^ { 2 } = 29.0 , \quad \sum y = 62.4 , \quad \sum y ^ { 2 } = 76.8 .$$ Dev claims that the mean volume of juice in bottles filled by machine \(Y\) is greater than the mean volume of juice in bottles filled by machine \(X\). A test at the \(\alpha \%\) significance level suggests that there is sufficient evidence to support Dev's claim. Find the set of possible values of \(\alpha\). \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-10_2717_33_109_2014} \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-11_2726_35_97_20}
CAIE Further Paper 4 2024 November Q6
9 marks Standard +0.3
6 A sports college keeps records of the times taken by students to run one lap of a running track. The population median time taken is 51.0 seconds. After a month of intensive training, a random sample of 22 new students run one lap of the track, giving times, in seconds, as follows.
51.352.053.449.249.351.152.247.2
53.048.549.450.350.851.649.152.3
51.852.447.948.950.651.9
It is claimed that the intensive training has led to a decrease in the median time taken to run one lap of the track. Carry out a Wilcoxon signed-rank test, at the \(5 \%\) significance level, to test whether there is sufficient evidence to support the claim. \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-13_2726_35_97_20}
If you use the following page to complete the answer to any question, the question number must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-14_2715_33_109_2012}
CAIE Further Paper 4 2024 November Q2
9 marks Standard +0.3
2 A school with a large number of students is updating its logo. Each student has designed a new logo and two teachers have each awarded a mark out of 50 for each logo. The marks awarded to a random sample of 12 students are shown in the following table.
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)\(K\)\(L\)
Teacher 1363840362234454448352830
Teacher 2384232413241425036444241
One of the students claims that Teacher 2 is awarding higher marks than Teacher 1.
  1. Carry out a Wilcoxon matched-pairs signed-rank test, at the \(5 \%\) significance level, to test whether the data supports the claim. \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-04_2715_38_109_2010} \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-05_2716_29_107_22} It was later discovered that Teacher 1 had entered her mark for student \(C\) incorrectly. Her intended mark was 24 not 40 . This was corrected.
  2. Determine whether this correction affects the conclusion of the test carried out in part (a).
CAIE Further Paper 4 2024 November Q3
8 marks Standard +0.3
3 A statistician believes that the number of telephone calls received by an advice centre in a 10 -minute interval can be modelled by the Poisson distribution \(\mathrm { Po } ( 1.9 )\). The number of calls received in a randomly chosen 10-minute interval was recorded on each of 100 days. The results are summarised in the table, together with some of the expected frequencies corresponding to the distribution \(\operatorname { Po } ( 1.9 )\).
Number of calls0123456 or more
Observed frequency101835211141
Expected frequency14.95728.41826.9971.322
  1. Complete the table.
  2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to determine whether the statistician's belief is reasonable. \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-07_2726_35_97_20}
CAIE Further Paper 4 2024 November Q4
10 marks Standard +0.3
4 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} k x ^ { 3 } & 0 \leqslant x < 1 \\ k ( 5 - x ) & 1 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Sketch the graph of f.
  2. Show that \(k = \frac { 4 } { 33 }\). \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-09_2725_35_99_20}
  3. Find the cumulative distribution function of \(X\).
  4. Find the median value of \(X\).
CAIE Further Paper 4 2024 November Q5
9 marks Standard +0.8
5 Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 3 }\) and the third coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 5 }\). The random variable \(X\) is the number of heads that Nikita obtains when he throws all three coins at the same time.
  1. Find the probability generating function of \(X\).
    Rajesh has two fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6. The random variable \(Y\) is the number of 4 s that Rajesh obtains when he throws the two dice. The random variable \(Z\) is the sum of the number of heads obtained by Nikita and the number of 4 s obtained by Rajesh.
  2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
    \includegraphics[max width=\textwidth, alt={}]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-10_444_33_106_2013}\includegraphics[max width=\textwidth, alt={}]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-10_443_33_675_2013}\includegraphics[max width=\textwidth, alt={}]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-10_440_33_1247_2013}\includegraphics[max width=\textwidth, alt={}]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-10_441_33_1816_2013}\includegraphics[max width=\textwidth, alt={}]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-10_443_31_2385_2015}
  3. Use your answer to part (b) to find \(\mathrm { E } ( Z )\).
CAIE Further Paper 4 2024 November Q6
8 marks Standard +0.3
6 Ansal is investigating the wingspans of Monarch butterflies in two different regions, \(X\) and \(Y\). He takes a random sample of 8 Monarch butterflies from region \(X\) and records their wingspans, \(x \mathrm {~cm}\). His results are as follows. $$\begin{array} { l l l l l l l l } 8.2 & 7.0 & 7.3 & 8.8 & 7.8 & 8.5 & 9.2 & 7.4 \end{array}$$ Ansal also takes a random sample of 9 Monarch butterflies from region \(Y\) and records their wingspans, \(y \mathrm {~cm}\). His results are summarised as follows. $$\sum y = 71.10 \quad \sum y ^ { 2 } = 567.13$$ Ansal suspects that the mean wingspan of Monarch butterflies from region \(X\) is greater than the mean wingspan of Monarch butterflies from region \(Y\). It is known that the wingspans of Monarch butterflies in regions \(X\) and \(Y\) are normally distributed with equal population variances. Test, at the 10\% significance level, whether Ansal's suspicion is supported by the data. \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-12_2715_44_110_2006} \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-13_2726_35_97_20}
If you use the following page to complete the answer to any question, the question number must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-14_2714_38_109_2010}
CAIE Further Paper 4 2020 Specimen Q1
7 marks Moderate -0.5
1
  1. State briefly the circumstances under which a non-parametric test of significance should be used rather than a parametric test. The level of pollution in a river was measured at 12 randomly chosen locations. The results, in suitable units, are shown below, where higher values represent greater pollution.
    5.625.736.556.816.105.755.876.475.866.266.995.91
  2. Use a Wilcoxon signed-rank test to test whether the average pollution level in the river is more than 6.00. Use a \(5\%\) significance level.
    [0pt] [6]
CAIE Further Paper 4 2020 Specimen Q2
7 marks Challenging +1.2
2 Each of 200 identically biased dice is thrown repeatedly until an even number is obtained. The number of throws needed is recorded and the results are summarised in the following table.
Number of throws123456\(\geqslant 7\)
Frequency12643223510
Carry out a goodness of fit test, at the \(5\%\) significance level, to test whether \(\operatorname{Geo}(0.6)\) is a satisfactory model for the data.
CAIE Further Paper 4 2020 Specimen Q3
8 marks Standard +0.3
3 Employees at a particular company have been working seven hours each day, from 9 am to 4 pm. To try to reduce absence, the company decides to introduce 'flexi-time' and allow employees to work their seven hours each day at any time between 7 am and 9 pm. For a random sample of 10 employees, the numbers of hours of absence in the year before and the year after the introduction of flexi-time are given in the following table.
Employee\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
Before4235967420578451460
After34321007231261351400
Test, at the \(10\%\) significance level, whether the population mean number of hours of absence has decreased following the introduction of flexi-time, stating any assumption that you make.
CAIE Further Paper 4 2020 Specimen Q4
7 marks Standard +0.8
4 The number, \(x\), of a certain type of sea shell was counted at 60 randomly chosen sites, each one metre square, along the coastline in country \(A\). The number, \(y\), of the same type of sea shell was counted at 50 randomly chosen sites, each one metre square, along the coastline in country \(B\). The results are summarised as follows, where \(\bar{x}\) and \(\bar{y}\) denote the sample means of \(x\) and \(y\) respectively. $$\bar{x} = 29.2 \quad \Sigma(x - \bar{x})^{2} = 4341.6 \quad \bar{y} = 24.4 \quad \Sigma(y - \bar{y})^{2} = 3732.0$$ Find a \(95\%\) confidence interval for the difference between the mean number of sea shells, per square metre, on the coastlines in country \(A\) and in country \(B\).
CAIE Further Paper 4 2020 Specimen Q5
8 marks Standard +0.3
5 The continuous random variable \(X\) has probability density function f given by $$f(x) = \begin{cases} 0 & x < 0 \\ \frac{6}{5} x & 0 \leqslant x \leqslant 1 \\ \frac{6}{5} x^{-4} & x > 1 \end{cases}$$
  1. Find \(\mathrm{P}(X > 1)\).
  2. Find the median value of \(X\).
  3. Given that \(\mathrm{E}(X) = 1\), find the variance of \(X\).
  4. Find \(\mathrm{E}(\sqrt{X})\).
CAIE Further Paper 4 2020 Specimen Q6
13 marks Standard +0.3
6 Aisha has a bag containing 3 red balls and 3 white balls. She selects a ball at random, notes its colour and returns it to the bag; the same process is repeated twice more. The number of red balls selected by Aisha is denoted by \(X\).
  1. Find the probability generating function \(\mathrm{G}_{X}(t)\) of \(X\).
Basant also has a bag containing 3 red balls and 3 white balls. He selects three balls at random, without replacement, from his bag. The number of red balls selected by Basant is denoted by \(Y\).
  1. Find the probability generating function \(\mathrm{G}_{Y}(t)\) of \(Y\).
The random variable \(Z\) is the total number of red balls selected by Aisha and Basant.
  1. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
  2. Use the probability generating function of \(Z\) to find \(\mathrm{E}(Z)\) and \(\operatorname{Var}(Z)\).