Questions — Edexcel S1 (574 questions)

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Edexcel S1 2020 June Q1
  1. The discrete random variable \(X\) takes the values \(- 1,2,3,4\) and 7 only.
Given that $$\mathrm { P } ( X = x ) = \frac { 8 - x } { k } \text { for } x = - 1,2,3,4 \text { and } 7$$ find the value of \(\mathrm { E } ( X )\)
Edexcel S1 2020 June Q2
  1. In a school canteen, students can choose from a main course of meat ( \(M\) ), fish ( \(F\) ) or vegetarian ( \(V\) ). They can then choose a drink of either water ( \(W\) ) or juice ( \(J\) ).
The partially completed tree diagram, where \(p\) and \(q\) are probabilities, shows the probabilities of these choices for a randomly selected student. \section*{Drink} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Main course} \includegraphics[alt={},max width=\textwidth]{81d5e460-9559-4d25-aa08-6440559aec83-04_783_1013_593_463}
\end{figure}
  1. Complete the tree diagram, giving your answers in terms of \(p\) and \(q\) where appropriate.
  2. Find an expression, in terms of \(p\) and \(q\), for the probability that a randomly selected student chooses water to drink. The events "choosing a vegetarian main course" and "choosing water to drink" are independent.
  3. Find a linear equation in terms of \(p\) and \(q\). A student who has chosen juice to drink is selected at random. The probability that they chose fish for their main course is \(\frac { 7 } { 30 }\)
  4. Find the value of \(p\) and the value of \(q\). The canteen manager claims that students who choose water to drink are most likely to choose a fish main course.
  5. State, showing your working clearly, whether or not the manager's claim is correct.
Edexcel S1 2020 June Q3
3. The distance achieved in a long jump competition by students at a school is normally Students who achieve a distance greater than 4.3 metres receive a medal.
  1. Find the proportion of students who receive medals. The school wishes to give a certificate of achievement or a medal to the \(80 \%\) of students who achieve a distance of at least \(d\) metres.
  2. Find the value of \(d\). Of those who received medals, the \(\frac { 1 } { 3 }\) who jump the furthest will receive gold medals.
  3. Find the shortest distance, \(g\) metres, that must be achieved to receive a gold medal. A journalist from the local newspaper interviews a randomly selected group of 3 medal winners.
  4. Find the exact probability that there is at least one gold medal winner in the group.
    \includegraphics[max width=\textwidth, alt={}, center]{81d5e460-9559-4d25-aa08-6440559aec83-08_79_1153_233_251} Students who achieve a distance greater than 4.3 metres receive a medal.
  5. Find the proportion of students who receive medals.
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel S1 2020 June Q4
4. A group of students took some tests. A teacher is analysing the average mark for each student. Each student obtained a different average mark. For these average marks, the lower quartile is 24 , the median is 30 and the interquartile range (IQR) is 10
The three lowest average marks are 8, 10 and 15.5 and the three highest average marks are 45, 52.5 and 56 The teacher defines an outlier to be a value that is either
more than \(1.5 \times\) IQR below the lower quartile or
more than \(1.5 \times\) IQR above the upper quartile
  1. Determine any outliers in these data.
  2. On the grid below draw a box plot for these data, indicating clearly any outliers.
    \includegraphics[max width=\textwidth, alt={}, center]{81d5e460-9559-4d25-aa08-6440559aec83-12_350_1223_1128_370}
  3. Use the quartiles to describe the skewness of these data. Give a reason for your answer. Two more students also took the tests. Their average marks, which were both less than 45, are added to the data and the box plot redrawn. The median and the upper quartile are the same but the lower quartile is now 26
  4. Redraw the box plot on the grid below.
    (3)
    \includegraphics[max width=\textwidth, alt={}, center]{81d5e460-9559-4d25-aa08-6440559aec83-12_350_1221_2106_367}
  5. Give ranges of values within which each of these students' average marks must lie. Turn over for spare grids if you need to redraw your answer for part (b) or part (d).
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
    \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Copy of grid for part (b)} \includegraphics[alt={},max width=\textwidth]{81d5e460-9559-4d25-aa08-6440559aec83-15_356_1226_1726_367}
    \end{figure} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Copy of grid for part (d)} \includegraphics[alt={},max width=\textwidth]{81d5e460-9559-4d25-aa08-6440559aec83-15_353_1226_2240_367}
    \end{figure}
Edexcel S1 2020 June Q5
  1. A large company rents shops in different parts of the country. A random sample of 10 shops was taken and the floor area, \(x\) in \(10 \mathrm {~m} ^ { 2 }\), and the annual rent, \(y\) in thousands of dollars, were recorded.
    The data are summarised by the following statistics
$$\sum x = 900 \quad \sum x ^ { 2 } = 84818 \quad \sum y = 183 \quad \sum y ^ { 2 } = 3434$$ and the regression line of \(y\) on \(x\) has equation \(y = 6.066 + 0.136 x\)
  1. Use the regression line to estimate the annual rent in dollars for a shop with a floor area of \(800 \mathrm {~m} ^ { 2 }\)
  2. Find \(\mathrm { S } _ { y y }\) and \(\mathrm { S } _ { x x }\)
  3. Find the product moment correlation coefficient between \(y\) and \(x\). An 11th shop is added to the sample. The floor area is \(900 \mathrm {~m} ^ { 2 }\) and the annual rent is 15000 dollars.
  4. Use the formula \(\mathrm { S } _ { x y } = \sum ( x - \bar { x } ) ( y - \bar { y } )\) to show that the value of \(\mathrm { S } _ { x y }\) for the 11 shops will be the same as it was for the original 10 shops.
  5. Find the new equation of the regression line of \(y\) on \(x\) for the 11 shops. The company is considering renting a larger shop with area of \(3000 \mathrm {~m} ^ { 2 }\)
  6. Comment on the suitability of using the new regression line to estimate the annual rent. Give a reason for your answer.
Edexcel S1 2020 June Q6
6. The random variable \(A\) represents the score when a spinner is spun. The probability distribution for \(A\) is given in the following table.
\(a\)1457
\(\mathrm { P } ( A = a )\)0.400.200.250.15
  1. Show that \(\mathrm { E } ( A ) = 3.5\)
  2. Find \(\operatorname { Var } ( A )\) The random variable \(B\) represents the score on a 4 -sided die. The probability distribution for \(B\) is given in the following table where \(k\) is a positive integer.
    \(b\)134\(k\)
    \(\mathrm { P } ( B = b )\)0.250.250.250.25
  3. Write down the name of the probability distribution of \(B\).
  4. Given that \(\mathrm { E } ( B ) = \mathrm { E } ( A )\) state, giving a reason, the value of \(k\). The random variable \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) Sam and Tim are playing a game with the spinner and the die. They each spin the spinner once to obtain their value of \(A\) and each roll the die once to obtain their value of \(B\).
    Their value of \(A\) is taken as their value of \(\mu\) and their value of \(B\) is taken as their value of \(\sigma\). The person with the larger value of \(\mathrm { P } ( X > 3.5 )\) is the winner.
  5. Given that Sam obtained values of \(a = 4\) and \(b = 3\) and Tim obtained \(b = 4\) find, giving a reason, the probability that Tim wins.
  6. Find the largest value of \(\mathrm { P } ( X > 3.5 )\) achievable in this game.
  7. Find the probability of achieving this value. \includegraphics[max width=\textwidth, alt={}, center]{81d5e460-9559-4d25-aa08-6440559aec83-21_2255_50_314_34}
Edexcel S1 2021 June Q1
  1. There are 7 red counters, 3 blue counters and 2 yellow counters in a bag. Gina selects a counter at random from the bag and keeps it. If the counter is yellow she does not select any more counters. If the counter is not yellow she randomly selects a second counter from the bag.
    1. Complete the tree diagram.
    First Counter
    Second Counter
    \includegraphics[max width=\textwidth, alt={}, center]{a439724e-b570-434d-bf75-de2b50915042-02_1147_1081_603_397} Given that Gina has selected a yellow counter,
  2. find the probability that she has 2 counters.
Edexcel S1 2021 June Q2
2. In the Venn diagram below, \(A , B\) and \(C\) are events and \(p , q , r\) and \(s\) are probabilities. The events \(A\) and \(C\) are independent and \(\mathrm { P } ( A ) = 0.65\)
\includegraphics[max width=\textwidth, alt={}, center]{a439724e-b570-434d-bf75-de2b50915042-04_373_815_397_568}
  1. State which two of the events \(A\), \(B\) and \(C\) are mutually exclusive.
  2. Find the value of \(r\) and the value of \(s\). The events ( \(A \cap C ^ { \prime }\) ) and ( \(B \cup C\) ) are also independent.
  3. Find the exact value of \(p\) and the exact value of \(q\). Give your answers as fractions.
Edexcel S1 2021 June Q3
  1. A random sample of 100 carrots is taken from a farm and their lengths, \(L \mathrm {~cm}\), recorded. The data are summarised in the following table.
Length, \(L\) cmFrequency, fClass mid point, \(\boldsymbol { x } \mathbf { c m }\)
\(5 \leqslant L < 8\)56.5
\(8 \leqslant L < 10\)139
\(10 \leqslant L < 12\)1611
\(12 \leqslant L < 15\)2513.5
\(15 \leqslant L < 20\)3017.5
\(20 \leqslant L < 28\)1124
A histogram is drawn to represent these data.
The bar representing the class \(5 \leqslant L < 8\) is 1.5 cm wide and 1 cm high.
  1. Find the width and height of the bar representing the class \(15 \leqslant L < 20\)
  2. Use linear interpolation to estimate the median length of these carrots.
  3. Estimate
    1. the mean length of these carrots,
    2. the standard deviation of the lengths of these carrots. A supermarket will only buy carrots with length between 9 cm and 22 cm .
  4. Estimate the proportion of carrots from the farm that the supermarket will buy. Any carrots that the supermarket does not buy are sold as animal feed. The farm makes a profit of 2.2 pence on each carrot sold to the supermarket, a profit of 0.8 pence on each carrot longer than 22 cm and a loss of 1.2 pence on each carrot shorter than 9 cm .
  5. Find an estimate of the mean profit per carrot made by the farm.
Edexcel S1 2021 June Q4
  1. Kris works in the mailroom of a large company and is responsible for all the letters sent by the company. The weights of letters sent by the company, \(W\) grams, have a normal distribution with mean 165 g and standard deviation 35 g .
    1. Estimate the proportion of letters sent by the company that weigh less than 120 g .
    Kris splits the letters to be sent into 3 categories: heavy, medium and light, with \(\frac { 1 } { 3 }\) of the letters in each category.
  2. Find the weight limits that determine medium letters. A heavy letter is chosen at random.
  3. Find the probability that this letter weighs less than 200 g . Kris chooses a random sample of 3 letters from those in the mailroom one day.
  4. Find the probability that there is one letter in each of the 3 categories.
Edexcel S1 2021 June Q5
  1. The discrete random variable \(X\) has the following probability distribution
\(x\)- 2- 1014
\(\mathrm { P } ( X = x )\)\(a\)\(b\)\(c\)\(b\)\(a\)
Given that \(\mathrm { E } ( X ) = 0.5\)
  1. find the value of \(a\). Given also that \(\operatorname { Var } ( X ) = 5.01\)
  2. find the value of \(b\) and the value of \(c\). The random variable \(Y = 5 - 8 X\)
  3. Find
    1. \(\mathrm { E } ( Y )\)
    2. \(\operatorname { Var } ( Y )\)
  4. Find \(\mathrm { P } \left( 4 X ^ { 2 } > Y \right)\)
Edexcel S1 2021 June Q6
  1. Two economics students, Andi and Behrouz, are studying some data relating to unemployment, \(x \%\), and increase in wages, \(y \%\), for a European country. The least squares regression line of \(y\) on \(x\) has equation
$$y = 3.684 - 0.3242 x$$ and $$\sum y = 23.7 \quad \sum y ^ { 2 } = 42.63 \quad \sum x ^ { 2 } = 756.81 \quad n = 16$$
  1. Show that \(\mathrm { S } _ { y y } = 7.524375\)
  2. Find \(\mathrm { S } _ { x x }\)
  3. Find the product moment correlation coefficient between \(x\) and \(y\). Behrouz claims that, assuming the model is valid, the data show that when unemployment is 2\% wages increase at over 3\%
  4. Explain how Behrouz could have come to this conclusion. Andi uses the formula $$\text { range } = \text { mean } \pm 3 \times \text { standard deviation }$$ to estimate the range of values for \(x\).
  5. Find estimates of the minimum value and the maximum value of \(x\) in these data using Andi's formula.
  6. Comment, giving a reason, on the reliability of Behrouz's claim. Andi suggests using the regression line with equation \(y = 3.684 - 0.3242 x\) to estimate unemployment when wages are increasing at \(2 \%\)
  7. Comment, giving a reason, on Andi's suggestion.
    \includegraphics[max width=\textwidth, alt={}]{a439724e-b570-434d-bf75-de2b50915042-20_2647_1835_118_116}
Edexcel S1 2022 June Q1
  1. The company Seafield requires contractors to record the number of hours they work each week. A random sample of 38 weeks is taken and the number of hours worked per week by contractor Kiana is summarised in the stem and leaf diagram below.
StemLeaf
144455566999(11)
212233444\(w\)9(10)
32344567779(10)
41123(4)
519(2)
64(1)
Key : 3|2 means 32 The quartiles for this distribution are summarised in the table below.
\(Q _ { 1 }\)\(Q _ { 2 }\)\(Q _ { 3 }\)
\(x\)26\(y\)
  1. Find the values of \(w , x\) and \(y\) Kiana is looking for outliers in the data. She decides to classify as outliers any observations greater than $$Q _ { 3 } + 1.0 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
  2. Showing your working clearly, identify any outliers that Kiana finds.
  3. Draw a box plot for these data in the space provided on the grid opposite.
  4. Use the formula $$\text { skewness } = \frac { \left( Q _ { 3 } - Q _ { 2 } \right) - \left( Q _ { 2 } - Q _ { 1 } \right) } { \left( Q _ { 3 } - Q _ { 1 } \right) }$$ to find the skewness of these data. Give your answer to 2 significant figures. Kiana's new employer, Landacre, wishes to know the average number of hours per week she worked during her employment at Seafield to help calculate the cost of employing her.
  5. Explain why Landacre might prefer to know Kiana's mean, rather than median, number of hours worked per week. Turn over for a spare grid if you need to redraw your box plot.
Edexcel S1 2022 June Q2
  1. Stuart is investigating the relationship between Gross Domestic Product (GDP) and the size of the population for a particular country.
    He takes a random sample of 9 years and records the size of the population, \(t\) millions, and the GDP, \(g\) billion dollars for each of these years.
The data are summarised as $$n = 9 \quad \sum t = 7.87 \quad \sum g = 144.84 \quad \sum g ^ { 2 } = 3624.41 \quad S _ { t t } = 1.29 \quad S _ { t g } = 40.25$$
  1. Calculate the product moment correlation coefficient between \(t\) and \(g\)
  2. Give an interpretation of your product moment correlation coefficient.
  3. Find the equation of the least squares regression line of \(g\) on \(t\) in the form \(g = a + b t\)
  4. Give an interpretation of the value of \(b\) in your regression line.
    1. Use the regression line from part (c) to estimate the GDP, in billions of dollars, for a population of 7000000
    2. Comment on the reliability of your answer in part (i). Give a reason, in context, for your answer. Using the regression line from part (c), Stuart estimates that for a population increase of \(x\) million there will be an increase of 0.1 billion dollars in GDP.
  6. Find the value of \(x\)
Edexcel S1 2022 June Q3
  1. Gill buys a bag of logs to use in her stove. The lengths, \(l \mathrm {~cm}\), of the 88 logs in the bag are summarised in the table below.
Length \(( \boldsymbol { l } )\)Frequency \(( \boldsymbol { f } )\)
\(15 < l \leqslant 20\)19
\(20 < l \leqslant 25\)35
\(25 < l \leqslant 27\)16
\(27 < l \leqslant 30\)15
\(30 < l \leqslant 40\)3
A histogram is drawn to represent these data.
The bar representing logs with length \(27 < l \leqslant 30\) has a width of 1.5 cm and a height of 4 cm .
  1. Calculate the width and height of the bar representing log lengths of \(20 < l \leqslant 25\)
  2. Use linear interpolation to estimate the median of \(l\) The maximum length of log Gill can use in her stove is 26 cm .
    Gill estimates, using linear interpolation, that \(x\) logs from the bag will fit into her stove.
  3. Show that \(x = 62\) Gill randomly selects 4 logs from the bag.
  4. Using \(x = 62\), find the probability that all 4 logs will fit into her stove. The weights, \(W\) grams, of the logs in the bag are coded using \(y = 0.5 w - 255\) and summarised by $$n = 88 \quad \sum y = 924 \quad \sum y ^ { 2 } = 12862$$
  5. Calculate
    1. the mean of \(W\)
    2. the variance of \(W\)
Edexcel S1 2022 June Q4
  1. The events \(H\) and \(W\) are such that
$$\mathrm { P } ( H ) = \frac { 3 } { 8 } \quad \mathrm { P } ( H \cup W ) = \frac { 3 } { 4 }$$ Given that \(H\) and \(W\) are independent,
  1. show that \(\mathrm { P } ( W ) = \frac { 3 } { 5 }\) The event \(N\) is such that $$\mathrm { P } ( N ) = \frac { 1 } { 15 } \quad \mathrm { P } ( H \cap N ) = \mathrm { P } ( N )$$
  2. Find \(\mathrm { P } \left( N ^ { \prime } \mid H \right)\) Given that \(W\) and \(N\) are mutually exclusive,
  3. draw a Venn diagram to represent the events \(H , W\) and \(N\) giving the exact probabilities of each region in the Venn diagram.
Edexcel S1 2022 June Q5
  1. A red spinner is designed so that the score \(R\) is given by the following probability distribution.
\(r\)23456
\(\mathrm { P } ( R = r )\)0.250.30.150.10.2
  1. Show that \(\mathrm { E } \left( R ^ { 2 } \right) = 15.8\) Given also that \(\mathrm { E } ( R ) = 3.7\)
  2. find the standard deviation of \(R\), giving your answer to 2 decimal places. A yellow spinner is designed so that the score \(Y\) is given by the probability distribution in the table below. The cumulative distribution function \(\mathrm { F } ( y )\) is also given.
    \(y\)23456
    \(\mathrm { P } ( Y = y )\)0.10.20.1\(a\)\(b\)
    \(\mathrm {~F} ( y )\)0.10.30.4\(c\)\(d\)
  3. Write down the value of \(d\) Given that \(\mathrm { E } ( Y ) = 4.55\)
  4. find the value of \(c\) Pabel and Jessie play a game with these two spinners.
    Pabel uses the red spinner.
    Jessie uses the yellow spinner.
    They take turns to spin their spinner.
    The winner is the first person whose spinner lands on the number 2 and the game ends. Jessie spins her spinner first.
  5. Find the probability that Jessie wins on her second spin.
  6. Calculate the probability that, in a game, the score on Pabel's first spin is the same as the score on Jessie’s first spin.
Edexcel S1 2022 June Q6
  1. A manufacturer fills bottles with oil. The volume of oil in a bottle, \(V \mathrm { ml }\), is normally distributed with \(V \sim \mathrm {~N} \left( 100,2.5 ^ { 2 } \right)\)
    1. Find \(\mathrm { P } ( V > 104.9 )\)
    2. In a pack of 150 bottles, find the expected number of bottles containing more than 104.9 ml
    3. Find the value of \(v\), to 2 decimal places, such that \(\mathrm { P } ( V > v \mid V < 104.9 ) = 0.2801\)
Edexcel S1 2023 June Q1
  1. The histogram shows the distances, in km , that 274 people travel to work.
    \includegraphics[max width=\textwidth, alt={}, center]{b8ac20db-4237-4def-81aa-a3eecbeefbdd-02_1272_1582_296_175}
Given that 60 of these people travel between 10 km and 20 km to work, estimate
  1. the number of people who travel between 22 km and 45 km to work,
  2. the median distance travelled to work by these 274 people,
  3. the mean distance travelled to work by these 274 people.
Edexcel S1 2023 June Q2
  1. Two students, Olive and Shan, collect data on the weight, \(w\) grams, and the tail length, \(t \mathrm {~cm}\), of 15 mice.
Olive summarised the data as follows $$\mathrm { S } _ { t t } = 5.3173 \quad \sum w ^ { 2 } = 6089.12 \quad \sum t w = 2304.53 \quad \sum w = 297.8 \quad \sum t = 114.8$$
  1. Calculate the value of \(\mathrm { S } _ { t w }\) and the value of \(\mathrm { S } _ { w w }\)
  2. Calculate the value of the product moment correlation coefficient between \(w\) and \(t\)
  3. Show that the equation of the regression line of \(w\) on \(t\) can be written as $$w = - 16.7 + 4.77 t$$
  4. Give an interpretation of the gradient of the regression line.
  5. Explain why it would not be appropriate to use the regression line in part (c) to estimate the weight of a mouse with a tail length of 2 cm . Shan decided to code the data using \(x = t - 6\) and \(y = \frac { w } { 2 } - 5\)
  6. Write down the value of the product moment correlation coefficient between \(x\) and \(y\)
  7. Write down an equation of the regression line of \(y\) on \(x\) You do not need to simplify your equation.
Edexcel S1 2023 June Q3
  1. Jim records the length, \(l \mathrm {~mm}\), of 81 salmon. The data are coded using \(x = l - 600\) and the following summary statistics are obtained.
$$n = 81 \quad \sum x = 3711 \quad \sum x ^ { 2 } = 475181$$
  1. Find the mean length of these salmon.
  2. Find the variance of the lengths of these salmon. The weight, \(w\) grams, of each of the 81 salmon is recorded to the nearest gram. The recorded results for the 81 salmon are summarised in the box plot below.
    \includegraphics[max width=\textwidth, alt={}, center]{b8ac20db-4237-4def-81aa-a3eecbeefbdd-10_362_1479_849_296}
  3. Find the maximum number of salmon that have weights in the interval $$4600 < w \leqslant 7700$$ Raj says that the box plot is incorrect as Jim has not included outliers.
    For these data an outlier is defined as a value that is more than
    \(1.5 \times\) IQR above the upper quartile or \(1.5 \times\) IQR below the lower quartile
  4. Show that there are no outliers.
Edexcel S1 2023 June Q4
  1. A bag contains a large number of coloured counters. Each counter is labelled A, B or C
    \(30 \%\) of the counters are labelled A
    \(45 \%\) of the counters are labelled B
    The rest of the counters are labelled C
    It is known that
    2\% of the counters labelled A are red
    4\% of the counters labelled B are red
    6\% of the counters labelled C are red
    One counter is selected at random from the bag.
    1. Complete the tree diagram on the opposite page to illustrate this information.
    2. Calculate the probability that the counter is labelled A and is not red.
    3. Calculate the probability that the counter is red.
    4. Given that the counter is red, find the probability that it is labelled C
    \includegraphics[max width=\textwidth, alt={}, center]{b8ac20db-4237-4def-81aa-a3eecbeefbdd-15_1155_1000_285_456}
Edexcel S1 2023 June Q6
6\% of the counters labelled C are red
One counter is selected at random from the bag.
  1. Complete the tree diagram on the opposite page to illustrate this information.
  2. Calculate the probability that the counter is labelled A and is not red.
  3. Calculate the probability that the counter is red.
  4. Given that the counter is red, find the probability that it is labelled C \end{enumerate} \includegraphics[max width=\textwidth, alt={}, center]{b8ac20db-4237-4def-81aa-a3eecbeefbdd-15_1155_1000_285_456}
    5. A discrete random variable \(Y\) has probability function $$\mathrm { P } ( \mathrm { Y } = \mathrm { y } ) = \left\{ \begin{array} { c l } \mathrm { k } ( 3 - \mathrm { y } ) & y = 1,2
    \mathrm { k } \left( \mathrm { y } ^ { 2 } - 8 \right) & y = 3,4,5
    \mathrm { k } & y = 6
    0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a constant.
  5. Show that \(k = \frac { 1 } { 30 }\) Find the exact value of
  6. \(\mathrm { P } ( 1 < Y \leqslant 4 )\)
  7. \(\mathrm { E } ( Y )\) The random variable \(X = 15 - 2 Y\)
  8. Calculate \(\mathrm { P } ( Y \geqslant X )\)
  9. Calculate \(\operatorname { Var } ( X )\)
    1. Three events \(A , B\) and \(C\) are such that
    $$\mathrm { P } ( A ) = 0.1 \quad \mathrm { P } ( B \mid A ) = 0.3 \quad \mathrm { P } ( A \cup B ) = 0.25 \quad \mathrm { P } ( C ) = 0.5$$ Given that \(A\) and \(C\) are mutually exclusive
  10. find \(\mathrm { P } ( A \cup C )\)
  11. Show that \(\mathrm { P } ( B ) = 0.18\) Given also that \(B\) and \(C\) are independent,
  12. draw a Venn diagram to represent the events \(A , B\) and \(C\) and the probabilities associated with each region.
Edexcel S1 2023 June Q7
  1. A machine squeezes apples to extract their juice. The volume of juice, \(J \mathrm { ml }\), extracted from 1 kg of apples is modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\)
Given that \(\mu = 500\) and \(\sigma = 25\) use standardisation to
    1. show that \(\mathrm { P } ( J > 510 ) = 0.3446\)
    2. calculate the value of \(d\) such that \(\mathrm { P } ( J > d ) = 0.9192\) Zen randomly selects 5 bags each containing 1 kg of apples and records the volume of juice extracted from each bag of apples.
  2. Calculate the probability that each of the 5 bags of apples produce less than 510 ml of juice. Following adjustments to the machine, the volume of juice, \(R \mathrm { ml }\), extracted from 1 kg of apples is such that \(\mu = 520\) and \(\sigma = k\) Given that \(\mathrm { P } ( R < r ) = 0.15\) and \(\mathrm { P } ( R > 3 r - 800 ) = 0.005\)
  3. find the value of \(r\) and the value of \(k\)
Edexcel S1 2024 June Q1
  1. A researcher is investigating the growth of two types of tree, Birch and Maple. The height, to the nearest cm, a seedling grows in one year is recorded for 35 Birch trees and 32 Maple trees. The results are summarised in the back-to-back stem and leaf diagram below.
TotalsBirchMapleTotals
(2)98257789(5)
(8)9996531130266899(7)
(9)9887631114\(111 \boldsymbol { k } 78\)(6)
(9)77754321050123444(7)
(3)7656346(3)
(3)654707(2)
(1)5800(2)
Key: 5 | 6 | 3 means 65 cm for a Birch tree and 63 cm for a Maple tree
The median height that these Maple trees grow in one year is 45 cm .
  1. Find the value of \(\boldsymbol { k }\), used in the stem and leaf diagram.
  2. Find the lower quartile and the upper quartile of the height grown in one year for these Birch trees. The researcher defines an outlier as an observation that is $$\text { greater than } Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { or less than } Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
  3. Show that there is only one outlier amongst the Birch trees. The grid on page 3 shows a box plot for the heights that the Maple trees grow in one year.
  4. On the same grid draw a box plot for the heights that the Birch trees grow in one year.
  5. Comment on any difference in the distributions of the growth of these Birch trees and the growth of these Maple trees.
    State the values of any statistics you have used to support your comment. The researcher realises he has missed out 4 pieces of data for the Maple trees. The heights each seedling grows in one year, to the nearest cm, in ascending order, for these 4 Maple trees are \(27 \mathrm {~cm} , a \mathrm {~cm} , 48 \mathrm {~cm} , 2 a \mathrm {~cm}\). Given that there is no change to the box plot for the Maple trees given on page 3
  6. find the range of possible values for \(a\) Show your working clearly.
    \includegraphics[max width=\textwidth, alt={}]{ee0c7c12-84f3-479c-b36a-3357f8529a1c-03_1243_1659_1464_210}
    Only use this grid if you need to redraw your answer for part (d)
    \includegraphics[max width=\textwidth, alt={}, center]{ee0c7c12-84f3-479c-b36a-3357f8529a1c-05_1154_1643_1503_217}
    (Total for Question 1 is 13 marks)