Questions — Edexcel S1 (574 questions)

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Edexcel S1 2017 January Q6
  1. In a factory, a machine is used to fill bags of rice. The weights of bags of rice are modelled using a normal distribution with mean 250 g .
Only \(1 \%\) of the bags of rice weigh more than 256 g .
  1. Write down the percentage of bags of rice with weights between 244 g and 256 g .
  2. Find the standard deviation of the weights of the bags of rice. An inspection consists of selecting a bag of rice at random and checking if its weight is within 4 g of the mean. If the weight is more than 4 g away from the mean, then a second bag of rice is selected at random and checked. If the weight of each of the 2 bags of rice is more than 4 g away from the mean, then the machine is shut down.
  3. Find the probability that the machine is shut down following an inspection.
Edexcel S1 2017 January Q7
  1. The discrete random variable \(X\) can take only the values \(1,2,3\) and 4 . For these values, the probability function is given by
$$\mathrm { P } ( X = x ) = \frac { a x + b } { 60 } \quad \text { for } x = 1,2,3,4$$ where \(a\) and \(b\) are constants.
  1. Show that \(5 a + 2 b = 30\) Given that \(\mathrm { F } ( 3 ) = \frac { 13 } { 20 }\)
  2. find the value of \(a\) and the value of \(b\) Given also that \(Y = X ^ { 2 }\)
  3. find the cumulative distribution function of \(Y\)
Edexcel S1 2018 January Q1
  1. Two classes of students, class \(A\) and class \(B\), sat a test.
Class \(A\) has 10 students. Class \(B\) has 15 students. Each student achieved a score, \(x\), on the test and their scores are summarised in the table below.
\cline { 2 - 4 } \multicolumn{1}{c|}{}\(n\)\(\sum x\)\(\sum x ^ { 2 }\)
Class \(A\)1077059610
Class \(B\)15\(t\)58035
The mean score for Class \(A\) is 77 and the mean score for Class \(B\) is 61
  1. Find the value of \(t\)
  2. Calculate the variance of the test scores for each class. The highest score on the test was 95 and the lowest score was 45 These were each scored by students from the same class.
  3. State, with a reason, which class you believe they were from. The two classes are combined into one group of 25 students.
    1. Find the mean test score for all 25 students.
    2. Find the variance of the test scores for all 25 students. The teacher of class \(A\) later realises that he added up the test scores for his class incorrectly. Each student's test score in class \(A\) should be increased by 3
  5. Without further calculations, state, with a reason, the effect this will have on
    1. the variance of the test scores for class \(A\)
    2. the mean test score for all 25 students
    3. the variance of the test scores for all 25 students.
Edexcel S1 2018 January Q2
2. (a) Shade the region representing the event \(A \cup B ^ { \prime }\) on the Venn diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{01259350-0119-4500-a81b-bfa1b4234559-06_355_563_306_694} The two events \(C\) and \(D\) are mutually exclusive.
Given that \(\mathrm { P } ( C ) = \frac { 1 } { 5 }\) and \(\mathrm { P } ( D ) = \frac { 3 } { 10 }\) find
(b) (i) \(\quad \mathrm { P } ( C \cup D )\)
(ii) \(\mathrm { P } ( C \mid D )\) The two events \(F\) and \(G\) are independent.
Given that \(\mathrm { P } ( F ) = \frac { 1 } { 6 }\) and \(\mathrm { P } ( F \cup G ) = \frac { 3 } { 8 }\) find
(c) (i) \(\mathrm { P } ( G )\)
(ii) \(\mathrm { P } \left( F \mid G ^ { \prime } \right)\)
Edexcel S1 2018 January Q3
3. Martin is investigating the relationship between a person's daily caffeine consumption, \(c\) milligrams, and the amount of sleep they get, \(h\) hours, per night. He collected this information from 20 people and the results are summarised below. $$\begin{array} { c c } \sum c = 3660 \quad \sum h = 126 \quad \sum c ^ { 2 } = 973228
\sum c h = 20023.4 \quad S _ { c c } = 303448 \quad S _ { c h } = - 3034.6 \end{array}$$ Martin calculates the product moment correlation coefficient for these data and obtains - 0.833
  1. Give a reason why this value supports a linear relationship between \(c\) and \(h\) The amount of sleep per night is the response variable.
  2. Explain what you understand by the term 'response variable'. Martin says that for each additional 100 mg of caffeine consumed, the expected number of hours of sleep decreases by 1
  3. Determine, by calculation, whether or not the data support this statement.
  4. Use the data to calculate an estimate for the expected number of hours of sleep per night when no caffeine is consumed.
Edexcel S1 2018 January Q4
4. The discrete random variable \(X\) has probability distribution
\(x\)- 4- 3125
\(\mathrm { P } ( X = x )\)\(a\)\(b\)\(a\)\(b\)0.2
  1. Find \(\mathrm { E } ( X )\) in terms of \(a\) and \(b\) For this probability distribution, \(\operatorname { Var } ( X ) = \mathrm { E } \left( X ^ { 2 } \right)\)
    1. Write down the value of \(\mathrm { E } ( X )\)
    2. Find the value of \(a\) and the value of \(b\)
  3. Find \(\operatorname { Var } ( 1 - 3 X )\) Given that \(Y = 1 - X\), find
    1. \(\mathrm { P } ( Y < 0 )\)
    2. the largest possible value of \(k\) such that \(\mathrm { P } ( Y < k ) = 0.2\)
Edexcel S1 2018 January Q5
5. Franca is the manager of an accountancy firm. She is investigating the relationship between the salary, \(\pounds x\), and the length of commute, \(y\) minutes, for employees at the firm. She collected this information from 9 randomly selected employees. The salary of each employee was then coded using \(w = \frac { x - 20000 } { 1000 }\) The table shows the values of \(w\) and \(y\) for the 9 employees.
\(w\)688- 125153- 219
\(y\)455035652540507520
(You may use \(\sum w = 81 \quad \sum y = 405 \quad \sum w y = 2490 \quad S _ { w w } = 660 \quad S _ { y y } = 2500\) )
  1. Calculate the salary of the employee with \(w = - 2\)
  2. Show that, to 3 significant figures, the value of the product moment correlation coefficient between \(w\) and \(y\) is - 0.899
  3. State, giving a reason, the value of the product moment correlation coefficient between \(x\) and \(y\) The least squares regression line of \(y\) on \(w\) is \(y = 60.75 - 1.75 w\)
  4. Find the equation of the least squares regression line of \(y\) on \(x\) giving your answer in the form \(y = a + b x\)
  5. Estimate the length of commute for an employee with a salary of \(\pounds 21000\) Franca uses the regression line to estimate the length of commute for employees with salaries between \(\pounds 25000\) and \(\pounds 40000\)
  6. State, giving a reason, whether or not these estimates are reliable.
Edexcel S1 2018 January Q6
  1. Anju has a bag that contains 5 socks of which 2 are blue.
Anju randomly selects socks from the bag, one sock at a time. She does not replace any socks but continues to select socks at random until she has both blue socks. The discrete random variable \(S\) represents the total number of socks that Anju has selected.
  1. Write down the value of \(\mathrm { P } ( S = 1 )\)
  2. Find \(\mathrm { P } ( S > 2 )\)
  3. Find \(\mathrm { P } ( S = 3 )\)
  4. Given that the second sock selected is blue, find the probability that Anju selects exactly 3 socks.
  5. Find \(\mathrm { P } ( S = 5 )\)
Edexcel S1 2018 January Q7
7. The weights, \(G\), of a particular breed of gorilla are normally distributed with mean 180 kg and standard deviation 15 kg .
  1. Find the proportion of these gorillas whose weights exceed 174 kg .
  2. Find, to 1 decimal place, the value of \(k\) such that \(\mathrm { P } ( k < G < 174 ) = 0.3196\) The weights, \(B\), of a particular breed of buffalo are normally distributed with mean 216 kg and standard deviation 30 kg . Given that \(\mathrm { P } ( G > w ) = \mathrm { P } ( B < w ) = p\)
    1. find the value of \(w\)
    2. find the value of \(p\) and standard deviation 15 kg .
  4. Find the proportion of these gorillas whose weights exceed 174 kg .
  5. Find, to 1 decimal place, the value of \(k\) such that \(\mathrm { P } ( k < G < 174 ) = 0.3196\)
    Leave blank
    Q7

    \hline &
    \hline \end{tabular}
Edexcel S1 2019 January Q1
  1. The Venn diagram shows the probability of a randomly selected student from a school being in the sets \(L , B\) and \(C\), where
    \(L\) represents the event that the student has instrumental music lessons
    \(B\) represents the event that the student plays in the school band
    \(C\) represents the event that the student sings in the school choir
    \(p , q , r\) and \(s\) are probabilities.
    \includegraphics[max width=\textwidth, alt={}, center]{d3f4450d-60eb-49b6-be1b-d2fcfad0451f-02_504_750_735_598}
    1. Select a pair of mutually exclusive events from \(L , B\) and \(C\).
    Given that \(\mathrm { P } ( L ) = 0.4 , \mathrm { P } ( B ) = 0.13 , \mathrm { P } ( C ) = 0.3\) and the events \(L\) and \(C\) are independent,
  2. find the value of \(p\),
  3. find the value of \(q\), the value of \(r\) and the value of \(s\). A student is selected at random from those who play in the school band or sing in the school choir.
  4. Find the exact probability that this student has instrumental music lessons.
Edexcel S1 2019 January Q2
2. The discrete random variable \(X\) has the following probability distribution.
\(x\)- 2- 1013
\(\mathrm { P } ( X = x )\)0.15\(a\)\(b\)\(a\)0.4
  1. Find \(\mathrm { E } ( X )\). Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 4.54\)
  2. find the value of \(a\) and the value of \(b\). The random variable \(Y = 3 - 2 X\)
  3. Find \(\operatorname { Var } ( Y )\).
Edexcel S1 2019 January Q3
  1. The weights of women boxers in a tournament are normally distributed with mean 64 kg and standard deviation 8 kg .
    1. Find the probability that a randomly chosen woman boxer in the tournament weighs less than 51 kg .
    In the tournament, women boxers who weigh less than 51 kg are classified as lightweight. Ren weighs 49 kg and she has a match against another randomly selected, lightweight woman boxer.
  2. Find the probability that Ren weighs less than the other boxer. In the tournament, women boxers who weigh more than \(H \mathrm {~kg}\) are classified as heavyweight. Given that \(10 \%\) of the women boxers in the tournament are classified as heavyweight,
  3. find the value of \(H\).
Edexcel S1 2019 January Q4
4. A group of 100 adults recorded the amount of time, \(t\) minutes, they spent exercising each day. Their results are summarised in the table below.
Time (t minutes)Frequency (f)Time midpoint (x)
\(0 \leqslant t < 15\)257.5
\(15 \leqslant t < 30\)1722.5
\(30 \leqslant t < 60\)2845
\(60 \leqslant t < 120\)2490
\(120 \leqslant t \leqslant 240\)6180
[You may use \(\sum \mathrm { f } x ^ { 2 } = 455\) 512.5]
A histogram is drawn to represent these data.
The bar representing the time \(0 \leqslant t < 15\) has width 0.5 cm and height 6 cm .
  1. Calculate the width and height of the bar representing a time of \(60 \leqslant t < 120\)
  2. Use linear interpolation to estimate the median time spent exercising by these adults each day.
  3. Find an estimate of the mean time spent exercising by these adults each day.
  4. Calculate an estimate for the standard deviation of these times.
  5. Describe, giving a reason, the skewness of these data. Further analysis of the above data revealed that 18 of the 25 adults in the \(0 \leqslant t < 15\) group took no exercise each day.
  6. State, giving a reason, what effect, if any, this new information would have on your answers to
    1. the estimate of the median in part (b),
    2. the estimate of the mean in part (c),
    3. the estimate of the standard deviation in part (d).
Edexcel S1 2019 January Q5
  1. Some children are playing a game involving throwing a ball into a bucket. Each child has 3 throws and the number of times the ball lands in the bucket, \(x\), is recorded. Their results are given in the table below.
\(x\)0123
Frequency1636244
  1. Find \(\bar { x }\)
    (1) Sandra decides to model the game by assuming that on each throw, the probability of the ball landing in the bucket is 0.4 for every child on every throw and that the throws are all independent. The random variable \(S\) represents the number of times the ball lands in the bucket for a randomly selected child.
  2. Find \(\mathrm { P } ( S = 2 )\)
  3. Complete the table below to show the probability distribution for \(S\).
    \(s\)0123
    \(\mathrm { P } ( S = s )\)0.4320.064
    Ting believes that the probability of the ball landing in the bucket is not the same for each throw. He suggests that the probability will increase with each throw and uses the model $$p _ { i } = 0.15 i + 0.10$$ where \(i = 1,2,3\) and \(p _ { i }\) is the probability that the \(i\) th throw of the ball, by any particular child, will land in the bucket.
    The random variable \(T\) represents the number of times the ball lands in the bucket for a randomly selected child using Ting’s model.
  4. Show that
    1. \(\mathrm { P } ( T = 3 ) = 0.055\)
    2. \(\mathrm { P } ( T = 1 ) = 0.45\)
      (5)
  5. Complete the table below to show the probability distribution for \(T\), stating the exact probabilities in each case.
    \(t\)0123
    \(\mathrm { P } ( T = t )\)0.450.055
  6. State, giving your reasons, whether Sandra's model or Ting's model is the more appropriate for modelling this game.
Edexcel S1 2019 January Q6
  1. Following some school examinations, Chetna is studying the results of the 16 students in her class. The mark for paper \(1 , x\), and the mark for paper \(2 , y\), for each student are summarised in the following statistics.
$$\bar { x } = 35.75 \quad \bar { y } = 25.75 \quad \sigma _ { x } = 7.79 \quad \sigma _ { y } = 11.91 \quad \sum x y = 15837$$
  1. Comment on the differences between the marks of the students on paper 1 and paper 2 Chetna decides to examine these data in more detail and plots the marks for each of the 16 students on the scatter diagram opposite.
    1. Explain why the circled point \(( 38,0 )\) is possibly an outlier.
    2. Suggest a possible reason for this result. Chetna decides to omit the data point \(( 38,0 )\) and examine the other 15 students' marks.
  3. Find the value of \(\bar { x }\) and the value of \(\bar { y }\) for these 15 students. For these 15 students
    1. explain why \(\sum x y\) is still 15837
    2. show that \(\mathrm { S } _ { x y } = 1169.8\) For these 15 students, Chetna calculates \(\mathrm { S } _ { x x } = 965.6\) and \(\mathrm { S } _ { y y } = 1561.7\) correct to 1 decimal place.
  5. Calculate the product moment correlation coefficient for these 15 students.
  6. Calculate the equation of the line of regression of \(y\) on \(x\) for these 15 students, giving your answer in the form \(y = a + b x\) The product moment correlation coefficient between \(x\) and \(y\) for all 16 students is 0.746
  7. Explain how your calculation in part (e) supports Chetna's decision to omit the point \(( 38,0 )\) before calculating the equation of the linear regression line.
    (1)
  8. Estimate the mark in the second paper for a student who scored 38 marks in the first paper.
    \includegraphics[max width=\textwidth, alt={}]{d3f4450d-60eb-49b6-be1b-d2fcfad0451f-17_1127_1146_301_406}
    \includegraphics[max width=\textwidth, alt={}]{d3f4450d-60eb-49b6-be1b-d2fcfad0451f-20_2630_1828_121_121}
Edexcel S1 2021 January Q1
  1. The Venn diagram shows the events \(A , B\) and \(C\) and their associated probabilities.
    \includegraphics[max width=\textwidth, alt={}, center]{4f034b9a-94c8-42f2-bd77-9adec277aba6-02_584_1061_296_445}
Find
  1. \(\mathrm { P } \left( B ^ { \prime } \right)\)
  2. \(\mathrm { P } ( A \cup C )\)
  3. \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\)
Edexcel S1 2021 January Q2
2. The stem and leaf diagram below shows the ages (in years) of the residents in a care home.
AgeKey: \(4 \mid 3\) is an age of 43
43\(( 1 )\)
54
6235688899\(( 1 )\)
711344666889\(( 9 )\)
80027889\(( 11 )\)
937
  1. Find the median age of the residents.
  2. Find the interquartile range (IQR) of the ages of the residents. An outlier is defined as a value that is either
    more than \(1.5 \times ( \mathrm { IQR } )\) below the lower quartile or more than \(1.5 \times ( \mathrm { IQR } )\) above the upper quartile.
  3. Determine any outliers in these data. Show clearly any calculations that you use.
  4. On the grid on page 5, draw a box plot to summarise these data.
    Ages
Edexcel S1 2021 January Q3
3. The weights of packages that arrive at a factory are normally distributed with a mean of 18 kg and a standard deviation of 5.4 kg
  1. Find the probability that a randomly selected package weighs less than 10 kg The heaviest 15\% of packages are moved around the factory by Jemima using a forklift truck.
  2. Find the weight, in kg , of the lightest of these packages that Jemima will move. One of the packages not moved by Jemima is selected at random.
  3. Find the probability that it weighs more than 18 kg A delivery of 4 packages is made to the factory. The weights of the packages are independent.
  4. Find the probability that exactly 2 of them will be moved by Jemima.
Edexcel S1 2021 January Q4
4. A spinner can land on the numbers \(10,12,14\) and 16 only and the probability of the spinner landing on each number is the same.
The random variable \(X\) represents the number that the spinner lands on when it is spun once.
  1. State the name of the probability distribution of \(X\).
    1. Write down the value of \(\mathrm { E } ( X )\)
    2. Find \(\operatorname { Var } ( X )\) A second spinner can land on the numbers 1, 2, 3, 4 and 5 only. The random variable \(Y\) represents the number that this spinner lands on when it is spun once. The probability distribution of \(Y\) is given in the table below
      \(y\)12345
      \(\mathrm { P } ( Y = y )\)\(\frac { 4 } { 30 }\)\(\frac { 9 } { 30 }\)\(\frac { 6 } { 30 }\)\(\frac { 5 } { 30 }\)\(\frac { 6 } { 30 }\)
  3. Find
    1. \(\mathrm { E } ( Y )\)
    2. \(\operatorname { Var } ( Y )\) The random variable \(W = a X + b\), where \(a\) and \(b\) are constants and \(a > 0\) Given that \(\mathrm { E } ( W ) = \mathrm { E } ( Y )\) and \(\operatorname { Var } ( W ) = \operatorname { Var } ( Y )\)
  4. find the value of \(a\) and the value of \(b\). Each of the two spinners is spun once.
  5. Find \(\mathrm { P } ( W = Y )\)
Edexcel S1 2021 January Q5
  1. A company director wants to introduce a performance-related pay structure for her managers. A random sample of 15 managers is taken and the annual salary, \(y\) in \(\pounds 1000\), was recorded for each manager. The director then calculated a performance score, \(x\), for each of these managers.
    The results are shown on the scatter diagram in Figure 1 on the next page.
    1. Describe the correlation between performance score and annual salary.
    The results are also summarised in the following statistics. $$\sum x = 465 \quad \sum y = 562 \quad \mathrm {~S} _ { x x } = 2492 \quad \sum y ^ { 2 } = 23140 \quad \sum x y = 19428$$
    1. Show that \(\mathrm { S } _ { x y } = 2006\)
    2. Find \(\mathrm { S } _ { y y }\)
  3. Find the product moment correlation coefficient between performance score and annual salary. The director believes that there is a linear relationship between performance score and annual salary.
  4. State, giving a reason, whether or not these data are consistent with the director's belief.
  5. Calculate the equation of the regression line of \(y\) on \(x\), in the form \(y = a + b x\) Give the value of \(a\) and the value of \(b\) to 3 significant figures.
  6. Give an interpretation of the value of \(b\).
  7. Plot your regression line on the scatter diagram in Figure 1 The director hears that one of the managers in the sample seems to be underperforming.
  8. On the scatter diagram, circle the point that best identifies this manager. The director decides to use this regression line for the new performance related pay structure.
    1. Estimate, to 3 significant figures, the new salary of a manager with a performance score of 30 \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{4f034b9a-94c8-42f2-bd77-9adec277aba6-15_1390_1408_299_187} \captionsetup{labelformat=empty} \caption{Figure 1}
      \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{4f034b9a-94c8-42f2-bd77-9adec277aba6-17_2654_99_115_9} Annual salary (£1000) \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Only use this scatter diagram if you need to redraw your line.} \includegraphics[alt={},max width=\textwidth]{4f034b9a-94c8-42f2-bd77-9adec277aba6-17_1378_1143_402_468}
      \end{figure}
Edexcel S1 2021 January Q6
  1. A disc of radius 1 cm is rolled onto a horizontal grid of rectangles so that the disc is equally likely to land anywhere on the grid. Each rectangle is 5 cm long and 3 cm wide. There are no gaps between the rectangles and the grid is sufficiently large so that no discs roll off the grid.
If the disc lands inside a rectangle without covering any part of the edges of the rectangle then a prize is won. By considering the possible positions for the centre of the disc,
  1. show that the probability of winning a prize on any particular roll is \(\frac { 1 } { 5 }\) A group of 15 students each roll the disc onto the grid twenty times and record the number of times, \(x\), that each student wins a prize. Their results are summarised as follows $$\sum x = 61 \quad \sum x ^ { 2 } = 295$$
  2. Find the standard deviation of the number of prizes won per student. A second group of 12 students each roll the disc onto the grid twenty times and the mean number of prizes won per student is 3.5 with a standard deviation of 2
  3. Find the mean and standard deviation of the number of prizes won per student for the whole group of 27 students. The 27 students also recorded the number of times that the disc covered a corner of a rectangle and estimated the probability to be 0.2216 (to 4 decimal places).
  4. Explain how this probability could be used to find an estimate for the value of \(\pi\) and state the value of your estimate.
Edexcel S1 2023 January Q1
  1. The histogram shows the times taken, \(t\) minutes, by each of 100 people to swim 500 metres.
    \includegraphics[max width=\textwidth, alt={}, center]{c316fa29-dedc-4890-bd82-31eb0bb819f9-02_986_1070_342_424}
    1. Use the histogram to complete the frequency table for the times taken by the 100 people to swim 500 metres.
    Time taken ( \(\boldsymbol { t }\) minutes)\(5 - 10\)\(10 - 14\)\(14 - 18\)\(18 - 25\)\(25 - 40\)
    Frequency ( \(\boldsymbol { f }\) )101624
  2. Estimate the number of people who took less than 16 minutes to swim 500 metres.
  3. Find an estimate for the mean time taken to swim 500 metres. Given that \(\sum f t ^ { 2 } = 41033\)
  4. find an estimate for the standard deviation of the times taken to swim 500 metres. Given that \(Q _ { 3 } = 23\)
  5. use linear interpolation to estimate the interquartile range of the times taken to swim 500 metres.
Edexcel S1 2023 January Q2
  1. Two bags, \(\boldsymbol { X }\) and \(\boldsymbol { Y }\), each contain green marbles (G) and blue marbles (B) only.
  • Bag \(\boldsymbol { X }\) contains 5 green marbles and 4 blue marbles
  • Bag \(\boldsymbol { Y }\) contains 6 green marbles and 5 blue marbles
A marble is selected at random from bag \(\boldsymbol { X }\) and placed in bag \(\boldsymbol { Y }\)
A second marble is selected at random from bag \(\boldsymbol { X }\) and placed in bag \(\boldsymbol { Y }\)
A third marble is then selected, this time from bag \(\boldsymbol { Y }\)
  1. Use this information to complete the tree diagram shown on page 7
  2. Find the probability that the 2 marbles selected from bag \(\boldsymbol { X }\) are of different colours.
  3. Find the probability that all 3 marbles selected are the same colour. Given that all three marbles selected are the same colour,
  4. find the probability that they are all green. 2nd Marble (from bag \(\boldsymbol { X }\) ) \section*{3rd Marble (from bag Y)} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{1st Marble (from bag \(\boldsymbol { X }\) )} \includegraphics[alt={},max width=\textwidth]{c316fa29-dedc-4890-bd82-31eb0bb819f9-07_1694_1312_484_310}
    \end{figure}
Edexcel S1 2023 January Q3
  1. The probability distribution of the discrete random variable \(X\) is given by
\(x\)234
\(\mathrm { P } ( X = x )\)\(a\)0.4\(0.6 - a\)
where \(a\) is a constant.
  1. Find, in terms of \(a , \mathrm { E } ( X )\)
  2. Find the range of the possible values of \(\mathrm { E } ( X )\) Given that \(\operatorname { Var } ( X ) = 0.56\)
  3. find the possible values of \(a\)
Edexcel S1 2023 January Q4
    1. In the Venn diagram below, \(A\) and \(B\) represent events and \(p , q , r\) and \(s\) are probabilities.
      \includegraphics[max width=\textwidth, alt={}, center]{c316fa29-dedc-4890-bd82-31eb0bb819f9-12_400_789_347_639}
$$\mathrm { P } ( A ) = \frac { 7 } { 25 } \quad \mathrm { P } ( B ) = \frac { 1 } { 5 } \quad \mathrm { P } \left[ \left( A \cap B ^ { \prime } \right) \cup \left( A ^ { \prime } \cap B \right) \right] = \frac { 8 } { 25 }$$
  1. Use algebra to show that \(2 p + 2 q + 2 r = \frac { 4 } { 5 }\)
  2. Find the value of \(p\), the value of \(q\), the value of \(r\) and the value of \(s\)
    (ii) Two events, \(C\) and \(D\), are such that $$\mathrm { P } ( C ) = \frac { x } { x + 5 } \quad \mathrm { P } ( D ) = \frac { 5 } { x }$$ where \(x\) is a positive constant.
    By considering \(\mathrm { P } ( C ) + \mathrm { P } ( D )\) show that \(C\) and \(D\) cannot be mutually exclusive.