Questions S1 (1967 questions)

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Edexcel S1 Q5
5. An antiques shop recorded the value of items stolen to the nearest pound during each week for a year giving the data in the table below.
Value of goods stolen (£)Number of weeks
0-19931
200-3996
400-5993
600-7994
800-9995
1000-19992
2000-29991
Letting \(x\) represent the mid-point of each group and using the coding \(y = \frac { x - 699.5 } { 200 }\),
  1. find \(\sum\) fy.
  2. estimate to the nearest pound the mean and standard deviation of the value of the goods stolen each week using your value for \(\sum f y\) and \(\sum f y ^ { 2 } = 424\).
    (6 marks)
    The median for these data is \(\pounds 82\).
  3. Explain why the manager of the shop might be reluctant to use either the mean or the median in summarising these data.
    (3 marks)
Edexcel S1 Q6
6. At the start of a gameshow there are 10 contestants of which 6 are female. In each round of the game, one contestant is eliminated. All of the contestants have the same chance of progressing to the next round each time.
  1. Show that the probability that the first two contestants to be eliminated are both male is \(\frac { 2 } { 15 }\).
  2. Find the probability that more females than males are eliminated in the first three rounds of the game.
  3. Given that the first contestant to be eliminated is male, find the probability that the next two contestants to be eliminated are both female.
    (3 marks)
Edexcel S1 Q7
7. A cyber-cafe recorded how long each user stayed during one day giving the following results.
Length of stay
(minutes)
\(0 -\)\(30 -\)\(60 -\)\(90 -\)\(120 -\)\(240 -\)\(360 -\)
Number of users153132231720
  1. Use linear interpolation to estimate the median and quartiles of these data. The results of a previous study had led to the suggestion that the length of time each user stays can be modelled by a normal distribution with a mean of 72 minutes and a standard deviation of 48 minutes.
  2. Find the median and quartiles that this model would predict.
  3. Comment on the suitability of the suggested model in the light of the new results.
CAIE S1 2006 June Q6
  1. How many teams play in only 1 match?
  2. How many teams play in exactly 2 matches?
  3. Draw up a frequency table for the numbers of matches which the teams play.
  4. Calculate the mean and variance of the numbers of matches which the teams play.
CAIE S1 2015 June Q4
(ii) Given that Nikita's mother does not like her present, find the probability that the present is a scarf.
CAIE S1 2014 November Q4
  1. Draw a fully labelled tree diagram to illustrate the various choices that Sharik can make until the computer indicates that he has answered the question correctly.
  2. The random variable \(X\) is the number of attempts that Sharik makes up to and including the one that the computer indicates is correct. Draw up the probability distribution table for \(X\) and find \(\mathrm { E } ( X )\).
    (a) The time, \(X\) hours, for which people sleep in one night has a normal distribution with mean 7.15 hours and standard deviation 0.88 hours.
  3. Find the probability that a randomly chosen person sleeps for less than 8 hours in a night.
  4. Find the value of \(q\) such that \(\mathrm { P } ( X < q ) = 0.75\).
    (b) The random variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\), where \(2 \sigma = 3 \mu\) and \(\mu \neq 0\). Find \(\mathrm { P } ( Y > 4 \mu )\).
OCR S1 2011 January Q5
  1. The number of free gifts that Jan receives in a week is denoted by \(X\). Name a suitable probability distribution with which to model \(X\), giving the value(s) of any parameter(s). State any assumption(s) necessary for the distribution to be a valid model. Assume now that your model is valid.
  2. Find
    (a) \(\mathrm { P } ( X \leqslant 2 )\),
    (b) \(\mathrm { P } ( X = 2 )\).
  3. Find the probability that, in the next 7 weeks, there are exactly 3 weeks in which Jan receives exactly 2 free gifts. 6
  4. The diagram shows 7 cards, each with a digit printed on it. The digits form a 7 -digit number. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
    3
    2
  5. 2
  6. 2
  7. 2
  8. \multirow[t]{21}{*}{3
  9. }
    10. \multirow[t]{4}{*}{3
  10. }
    11. 3
  11. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
OCR S1 2012 January Q7
  1. State a suitable distribution that can be used as a model for \(X\), giving the value(s) of any parameter(s). State also any necessary condition(s) for this distribution to be a good model. Use the distribution stated in part (i) to find
  2. \(\mathrm { P } ( X = 4 )\),
  3. \(\mathrm { P } ( X \geqslant 4 )\).
OCR MEI S1 2005 June Q5
  1. On the insert, complete the table giving the lowest common multiples of all pairs of integers between 1 and 6 .
    [0pt] [1]
    \multirow{2}{*}{}Second integer
    123456
    \multirow{6}{*}{First integer}1123456
    22264106
    336312156
    4441212
    551015
    666612
    Two fair dice are thrown and the lowest common multiple of the two scores is found.
  2. Use the table to find the probabilities of the following events.
    (A) The lowest common multiple is greater than 6 .
    (B) The lowest common multiple is a multiple of 5 .
    (C) The lowest common multiple is both greater than 6 and a multiple of 5 .
  3. Use your answers to part (ii) to show that the events "the lowest common multiple is greater than 6 " and "the lowest common multiple is a multiple of 5 " are not independent.
Edexcel S1 Q2
  1. Plot a scatter diagram showing these data. The student wanted to investigate further whether or not her data provided evidence of an increase in temperature in June each year. Using \(Y\) for the number of years since 1993 and \(T\) for the mean temperature, she calculated the following summary statistics. $$\Sigma Y = 28 , \quad \Sigma T = 182.5 , \quad \Sigma Y ^ { 2 } = 140 , \quad \Sigma T ^ { 2 } = 4173.93 , \quad \Sigma Y T = 644.7 .$$
  2. Calculate the product moment correlation coefficient for these data.
  3. Comment on your result in relation to the student's enquiry.
OCR MEI S1 Q3
  1. On die insert, complete the lable giving due lowest common multiples of all pairs of integers between 1 and 6 .
    Second integer
    \cline { 2 - 8 } \multicolumn{2}{|c|}{}123456
    \multirow{5}{*}{
    First
    integer
    }    		1123456
    \cline { 2 - 8 }22264106
    \cline { 2 - 8 }336312156
    \cline { 2 - 8 }4441212
    \cline { 2 - 8 }551015
    \cline { 2 - 8 }666612
    Two fair dice are thrown and the lowest common multiple of the two scores is found.
  2. Use the table to find the probabilities of the following events.
    (A) The lowest common multiple is greater than 6 .
    (B) The lowest common multiple is a multiple of 5 .
    (C) The lowest common multiple is both greater than 6 and a multiple of 5.
  3. Use your answers to part (ii) to show that the events "the lowest common multiple is greater than 6 " and "the lowest common multiple is a multiple of 5 " are not independent.
OCR MEI S1 2005 June Q6
6 Answer part (i) of this question on the insert provided. Mancaster Hockey Club invite prospective new players to take part in a series of three trial games. At the end of each game the performance of each player is assessed as pass or fail. Players who achieve a pass in all three games are invited to join the first team squad. Players who achieve a pass in two games are invited to join the second team squad. Players who fail in two games are asked to leave. This may happen after two games.
  • The probability of passing the first game is 0.9
  • Players who pass any game have probability 0.9 of passing the next game
  • Players who fail any game have probability 0.5 of failing the next game
    1. On the insert, complete the tree diagram which illustrates the information above.
      \includegraphics[max width=\textwidth, alt={}, center]{668963b4-994d-475a-a1c8-c3e3a252e4e6-4_691_1329_978_397}
    2. Find the probability that a randomly selected player
      (A) is invited to join the first team squad,
      (B) is invited to join the second team squad.
    3. Hence write down the probability that a randomly selected player is asked to leave.
    4. Find the probability that a randomly selected player is asked to leave after two games, given that the player is asked to leave.
Angela, Bryony and Shareen attend the trials at the same time. Assuming their performances are independent, find the probability that
  • at least one of the three is asked to leave,
  • they pass a total of 7 games between them.
    •
    OCR MEI S1 Q3
    3 Answer part (i) of this question on the insert provided. A taxi driver operates from a taxi rank at a main railway station in London. During one particular week he makes 120 journeys, the lengths of which are summarised in the table.
    Length
    \(( x\) miles \()\)
    		\(0 < x \leqslant 1\)\(1 < x \leqslant 2\)\(2 < x \leqslant 3\)\(3 < x \leqslant 4\)\(4 < x \leqslant 6\)\(6 < x \leqslant 10\)
    Number of
    journeys
    		3830211498
    1. On the insert, draw a cumulative frequency diagram to illustrate the data.
    2. Use your graph to estimate the median length of journey and the quartiles. Hence find the interquartile range.
    3. State the type of skewness of the distribution of the data.
    OCR MEI S1 Q4
    4 Answer part (i) of this question on the insert provided. Mancaster Hockey Club invite prospective new players to take part in a series of three trial games. At the end of each game the performance of each player is assessed as pass or fail. Players who achieve a pass in all three games are invited to join the first team squad. Players who achieve a pass in two games are invited to join the second team squad. Players who fail in two games are asked to leave. This may happen after two games.
    • The probability of passing the first game is 0.9
    • Players who pass any game have probability 0.9 of passing the next game
    • Players who fail any game have probability 0.5 of failing the next game
      1. On the insert, complete the tree diagram which illustrates the information above.
        \includegraphics[max width=\textwidth, alt={}, center]{64f25a40-d3bf-4212-b92e-655f980c702b-4_643_1239_942_417}
      2. Find the probability that a randomly selected player
        (A) is invited to join the first team squad,
        (B) is invited to join the second team squad.
      3. Hence write down the probability that a randomly selected player is asked to leave.
      4. Find the probability that a randomly selected player is asked to leave after two games, given that the player is asked to leave.
    Angela, Bryony and Shareen attend the trials at the same time. Assuming their performances are independent, find the probability that
  • at least one of the three is asked to leave,
  • they pass a total of 7 games between them.
    •
    OCR MEI S1 Q4
    4 Answer part (i) of this question on the insert provided. A taxi driver operates from a taxi rank at a main railway station in London. During one particular week he makes 120 journeys, the lengths of which are summarised in the table.
    Length
    \(( x\) miles \()\)
    		\(0 < x \leqslant 1\)\(1 < x \leqslant 2\)\(2 < x \leqslant 3\)\(3 < x \leqslant 4\)\(4 < x \leqslant 6\)\(6 < x \leqslant 10\)
    Number of
    journeys
    		3830211498
    1. On the insert, draw a cumulative frequency diagram to illustrate the data.
    2. Use your graph to estimate the median length of journey and the quartiles. Hence find the interquartile range.
    3. State the type of skewness of the distribution of the data.
    CAIE S1 2021 November Q1
    1 Each of the 180 students at a college plays exactly one of the piano, the guitar and the drums. The numbers of male and female students who play the piano, the guitar and the drums are given in the following table.
    PianoGuitarDrums
    Male254411
    Female423820
    A student at the college is chosen at random.
    1. Find the probability that the student plays the guitar.
    2. Find the probability that the student is male given that the student plays the drums.
    3. Determine whether the events 'the student plays the guitar' and 'the student is female' are independent, justifying your answer.
    CAIE S1 2021 November Q2
    2 A group of 6 people is to be chosen from 4 men and 11 women.
    1. In how many different ways can a group of 6 be chosen if it must contain exactly 1 man?
      Two of the 11 women are sisters Jane and Kate.
    2. In how many different ways can a group of 6 be chosen if Jane and Kate cannot both be in the group?
    CAIE S1 2021 November Q3
    3 A bag contains 5 yellow and 4 green marbles. Three marbles are selected at random from the bag, without replacement.
    1. Show that the probability that exactly one of the marbles is yellow is \(\frac { 5 } { 14 }\).
      The random variable \(X\) is the number of yellow marbles selected.
    2. Draw up the probability distribution table for \(X\).
    3. Find \(\mathrm { E } ( X )\).
    CAIE S1 2021 November Q4
    4
    1. In how many different ways can the 9 letters of the word TELESCOPE be arranged?
    2. In how many different ways can the 9 letters of the word TELESCOPE be arranged so that there are exactly two letters between the T and the C ?
    CAIE S1 2021 November Q5
    5 In a certain region, the probability that any given day in October is wet is 0.16 , independently of other days.
    1. Find the probability that, in a 10-day period in October, fewer than 3 days will be wet.
    2. Find the probability that the first wet day in October is 8 October.
    3. For 4 randomly chosen years, find the probability that in exactly 1 of these years the first wet day in October is 8 October.
    CAIE S1 2021 November Q6
    6 The times taken, in minutes, to complete a particular task by employees at a large company are normally distributed with mean 32.2 and standard deviation 9.6.
    1. Find the probability that a randomly chosen employee takes more than 28.6 minutes to complete the task.
    2. \(20 \%\) of employees take longer than \(t\) minutes to complete the task. Find the value of \(t\).
    3. Find the probability that the time taken to complete the task by a randomly chosen employee differs from the mean by less than 15.0 minutes.
    CAIE S1 2021 November Q8
    8
    *
    \end{tabular} & MATHEMATICS & 9709/52
    \hline 0 & Paper 5 Probability \& Statistics 1 & October/November 2021
    \hline \(\infty\) & & 1 hour 15 minutes
    \hline & You must answer on the question paper. &
    \hline & You will need: List of formulae (MF19) &
    \hline \end{tabular} \end{center} \section*{INSTRUCTIONS}
    • Answer all questions.
    • Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
    • Write your name, centre number and candidate number in the boxes at the top of the page.
    • Write your answer to each question in the space provided.
    • Do not use an erasable pen or correction fluid.
    • Do not write on any bar codes.
    • If additional space is needed, you should use the lined page at the end of this booklet; the question number or numbers must be clearly shown.
    • You should use a calculator where appropriate.
    • You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.
    • Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.
    \section*{INFORMATION}
    • The total mark for this paper is 50.
    • The number of marks for each question or part question is shown in brackets [ ].
    1 Each of the 180 students at a college plays exactly one of the piano, the guitar and the drums. The numbers of male and female students who play the piano, the guitar and the drums are given in the following table.
    PianoGuitarDrums
    Male254411
    Female423820
    A student at the college is chosen at random.
    1. Find the probability that the student plays the guitar.
    2. Find the probability that the student is male given that the student plays the drums.
    3. Determine whether the events 'the student plays the guitar' and 'the student is female' are independent, justifying your answer.
      2 A group of 6 people is to be chosen from 4 men and 11 women.
    4. In how many different ways can a group of 6 be chosen if it must contain exactly 1 man?
      Two of the 11 women are sisters Jane and Kate.
    5. In how many different ways can a group of 6 be chosen if Jane and Kate cannot both be in the group?
      3 A bag contains 5 yellow and 4 green marbles. Three marbles are selected at random from the bag, without replacement.
    6. Show that the probability that exactly one of the marbles is yellow is \(\frac { 5 } { 14 }\).
      The random variable \(X\) is the number of yellow marbles selected.
    7. Draw up the probability distribution table for \(X\).
    8. Find \(\mathrm { E } ( X )\).
      4
    9. In how many different ways can the 9 letters of the word TELESCOPE be arranged?
    10. In how many different ways can the 9 letters of the word TELESCOPE be arranged so that there are exactly two letters between the T and the C ?
      5 In a certain region, the probability that any given day in October is wet is 0.16 , independently of other days.
    11. Find the probability that, in a 10-day period in October, fewer than 3 days will be wet.
    12. Find the probability that the first wet day in October is 8 October.
    13. For 4 randomly chosen years, find the probability that in exactly 1 of these years the first wet day in October is 8 October.
      6 The times taken, in minutes, to complete a particular task by employees at a large company are normally distributed with mean 32.2 and standard deviation 9.6.
    14. Find the probability that a randomly chosen employee takes more than 28.6 minutes to complete the task.
    15. \(20 \%\) of employees take longer than \(t\) minutes to complete the task. Find the value of \(t\).
    16. Find the probability that the time taken to complete the task by a randomly chosen employee differs from the mean by less than 15.0 minutes.
      7 The distances, \(x \mathrm {~m}\), travelled to school by 140 children were recorded. The results are summarised in the table below.
      Distance, \(x \mathrm {~m}\)\(x \leqslant 200\)\(x \leqslant 300\)\(x \leqslant 500\)\(x \leqslant 900\)\(x \leqslant 1200\)\(x \leqslant 1600\)
      Cumulative frequency164688122134140
    17. On the grid, draw a cumulative frequency graph to represent these results.
      \includegraphics[max width=\textwidth, alt={}, center]{93ff111b-0267-4b4b-a41c-64c3307115af-10_1593_1593_701_306}
    18. Use your graph to estimate the interquartile range of the distances.
    19. Calculate estimates of the mean and standard deviation of the distances.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
    Edexcel S1 2022 January Q1
    1. A factory produces shoes.
    A quality control inspector at the factory checks a sample of 120 shoes for each of three types of defect. The Venn diagram represents the inspector's results. A represents the event that a shoe has defective stitching \(B\) represents the event that a shoe has defective colouring \(C\) represents the event that a shoe has defective soles
    \includegraphics[max width=\textwidth, alt={}, center]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-02_684_935_607_566} One of the shoes in the sample is selected at random.
    1. Find the probability that it does not have defective soles.
    2. Find \(\mathrm { P } \left( A \cap B \cap C ^ { \prime } \right)\)
    3. Find \(\mathrm { P } \left( A \cup B \cup C ^ { \prime } \right)\)
    4. Find the probability that the shoe has at most one type of defect.
    5. Given the selected shoe has at most one type of defect, find the probability it has defective stitching. The random variable \(X\) is the number of the events \(A , B , C\) that occur for a randomly selected shoe.
    6. Find \(\mathrm { E } ( X )\) \section*{This is a copy of the Venn diagram for this question.} \includegraphics[max width=\textwidth, alt={}, center]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-05_684_940_388_566}
    Edexcel S1 2022 January Q2
    2. Tom's car holds 50 litres of petrol when the fuel tank is full. For each of 10 journeys, each starting with 50 litres of petrol in the fuel tank, Tom records the distance travelled, \(d\) kilometres, and the amount of petrol used, \(p\) litres. The summary statistics for the 10 journeys are given below. $$\sum d = 1029 \quad \sum p = 50.8 \quad \sum d p = 5240.8 \quad \mathrm {~S} _ { d d } = 344.9 \quad \mathrm {~S} _ { p p } = 0.576$$
    1. Calculate the product moment correlation coefficient between \(d\) and \(p\) The amount of petrol remaining in the fuel tank for each journey, \(w\) litres, is recorded.
      1. Write down an equation for \(w\) in terms of \(p\)
      2. Hence, write down the value of the product moment correlation coefficient between \(w\) and \(p\)
    3. Write down the value of the product moment correlation coefficient between \(d\) and \(w\)
    Edexcel S1 2022 January Q3
    1. The stem and leaf diagram shows the number of deliveries made by Pat each day for 24 days
    \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Key: 10 \(\mathbf { 8 }\) represents 108 deliveries}
    1089(2)
    1103666889999(11)
    1245555558(8)
    13\(a\)\(b\)\(c\)(3)
    \end{table} where \(a\), \(b\) and \(c\) are positive integers with \(a < b < c\)
    An outlier is defined as any value greater than \(1.5 \times\) interquartile range above the upper quartile. Given that there is only one outlier for these data,
    1. show that \(c = 9\) The number of deliveries made by Pat each day is represented by \(d\)
      The data in the stem and leaf diagram are coded using $$x = d - 125$$ and the following summary statistics are obtained $$\sum x = - 96 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 1306$$
    2. Find the mean number of deliveries.
    3. Find the standard deviation of the number of deliveries. One of these 24 days is selected at random. The random variable \(D\) represents the number of deliveries made by Pat on this day. The random variable \(X = D - 125\)
    4. Find \(\mathrm { P } ( D > 118 \mid X < 0 )\)