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OCR S1 2009 January Q6
12 marks Standard +0.3
6 A test consists of 4 algebra questions, A, B, C and D, and 4 geometry questions, G, H, I and J.
The examiner plans to arrange all 8 questions in a random order, regardless of topic.
  1. (a) How many different arrangements are possible?
    (b) Find the probability that no two Algebra questions are next to each other and no two Geometry questions are next to each other. Later, the examiner decides that the questions should be arranged in two sections, Algebra followed by Geometry, with the questions in each section arranged in a random order.
  2. (a) How many different arrangements are possible?
    (b) Find the probability that questions A and H are next to each other.
    (c) Find the probability that questions B and J are separated by more than four other questions.
OCR S1 2009 January Q7
12 marks Moderate -0.8
7 At a factory that makes crockery the quality control department has found that \(10 \%\) of plates have minor faults. These are classed as 'seconds'. Plates are stored in batches of 12. The number of seconds in a batch is denoted by \(X\).
  1. State an appropriate distribution with which to model \(X\). Give the value(s) of any parameter(s) and state any assumptions required for the model to be valid. Assume now that your model is valid.
  2. Find
    (a) \(\mathrm { P } ( X = 3 )\),
    (b) \(\mathrm { P } ( X \geqslant 1 )\).
  3. A random sample of 4 batches is selected. Find the probability that the number of these batches that contain at least 1 second is fewer than 3 .
OCR S1 2009 January Q8
7 marks Moderate -0.3
8 A game uses an unbiased die with faces numbered 1 to 6 . The die is thrown once. If it shows 4 or 5 or 6 then this number is the final score. If it shows 1 or 2 or 3 then the die is thrown again and the final score is the sum of the numbers shown on the two throws.
  1. Find the probability that the final score is 4 .
  2. Given that the die is thrown only once, find the probability that the final score is 4 .
  3. Given that the die is thrown twice, find the probability that the final score is 4 .
OCR S1 2010 January Q1
9 marks Moderate -0.8
1 Andy makes repeated attempts to thread a needle. The number of attempts up to and including his first success is denoted by \(X\).
  1. State two conditions necessary for \(X\) to have a geometric distribution.
  2. Assuming that \(X\) has the distribution \(\operatorname { Geo } ( 0.3 )\), find
    (a) \(\mathrm { P } ( X = 5 )\),
    (b) \(\mathrm { P } ( X > 5 )\).
  3. Suggest a reason why one of the conditions you have given in part (i) might not be satisfied in this context. 240 people were asked to guess the length of a certain road. Each person gave their guess, \(l \mathrm {~km}\), correct to the nearest kilometre. The results are summarised below.
    \(l\)\(10 - 12\)\(13 - 15\)\(16 - 20\)\(21 - 30\)
    Frequency113206
  4. (a) Use appropriate formulae to calculate estimates of the mean and standard deviation of \(l\).
    (b) Explain why your answers are only estimates.
  5. A histogram is to be drawn to illustrate the data. Calculate the frequency density of the block for the 16-20 class.
  6. Explain which class contains the median value of \(l\).
  7. Later, the person whose guess was between 10 km and 12 km changed his guess to between 13 km and 15 km . Without calculation state whether the following will increase, decrease or remain the same:
    (a) the mean of \(l\),
    (b) the standard deviation of \(l\).
OCR S1 2010 January Q3
7 marks Moderate -0.8
3 The heights, \(h \mathrm {~m}\), and weights, \(m \mathrm {~kg}\), of five men were measured. The results are plotted on the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5c25d6cf-2c23-4b49-88fb-e4abe6c281e4-3_738_956_386_593} The results are summarised as follows. $$n = 5 \quad \Sigma h = 9.02 \quad \Sigma m = 377.7 \quad \Sigma h ^ { 2 } = 16.382 \quad \Sigma m ^ { 2 } = 28558.67 \quad \Sigma h m = 681.612$$
  1. Use the summarised data to calculate the value of the product moment correlation coefficient, \(r\).
  2. Comment on your value of \(r\) in relation to the diagram.
  3. It was decided to re-calculate the value of \(r\) after converting the heights to feet and the masses to pounds. State what effect, if any, this will have on the value of \(r\).
  4. One of the men had height 1.63 m and mass 78.4 kg . The data for this man were removed and the value of \(r\) was re-calculated using the original data for the remaining four men. State in general terms what effect, if any, this will have on the value of \(r\).
OCR S1 2010 January Q4
10 marks Moderate -0.8
4 A certain four-sided die is biased. The score, \(X\), on each throw is a random variable with probability distribution as shown in the table. Throws of the die are independent.
\(x\)0123
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 2 }\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 8 }\)\(\frac { 1 } { 8 }\)
  1. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\). The die is thrown 10 times.
  2. Find the probability that there are not more than 4 throws on which the score is 1 .
  3. Find the probability that there are exactly 4 throws on which the score is 2 .
OCR S1 2010 January Q5
6 marks Moderate -0.8
5 A washing-up bowl contains 6 spoons, 5 forks and 3 knives. Three of these 14 items are removed at random, without replacement. Find the probability that
  1. all three items are of different kinds,
  2. all three items are of the same kind.
OCR S1 2010 January Q6
7 marks Standard +0.3
6
  1. A student calculated the values of the product moment correlation coefficient, \(r\), and Spearman's rank correlation coefficient, \(r _ { s }\), for two sets of bivariate data, \(A\) and \(B\). His results are given below. $$\begin{array} { l l } A : & r = 0.9 \text { and } r _ { s } = 1 \\ B : & r = 1 \quad \text { and } r _ { s } = 0.9 \end{array}$$ With the aid of a diagram where appropriate, explain why the student's results for \(A\) could both be correct but his results for \(B\) cannot both be correct.
  2. An old research paper has been partially destroyed. The surviving part of the paper contains the following incomplete information about some bivariate data from an experiment. \includegraphics[max width=\textwidth, alt={}, center]{5c25d6cf-2c23-4b49-88fb-e4abe6c281e4-4_339_1200_1117_511} Calculate the missing constant at the end of the equation of the second regression line.
OCR S1 2010 January Q7
6 marks Easy -1.3
7 The table shows the numbers of male and female members of a vintage car club who own either a Jaguar or a Bentley. No member owns both makes of car.
\cline { 2 - 3 } \multicolumn{1}{c|}{}MaleFemale
Jaguar2515
Bentley128
One member is chosen at random from these 60 members.
  1. Given that this member is male, find the probability that he owns a Jaguar. Now two members are chosen at random from the 60 members. They are chosen one at a time, without replacement.
  2. Given that the first one of these members is female, find the probability that both own Jaguars.
OCR S1 2010 January Q8
7 marks Moderate -0.8
8 The five letters of the word NEVER are arranged in random order in a straight line.
  1. How many different orders of the letters are possible?
  2. In how many of the possible orders are the two Es next to each other?
  3. Find the probability that the first two letters in the order include exactly one letter E. \(9 R\) and \(S\) are independent random variables each having the distribution \(\operatorname { Geo } ( p )\).
  4. Find \(\mathrm { P } ( R = 1\) and \(S = 1 )\) in terms of \(p\).
  5. Show that \(\mathrm { P } ( R = 3\) and \(S = 3 ) = p ^ { 2 } q ^ { 4 }\), where \(q = 1 - p\).
  6. Use the formula for the sum to infinity of a geometric series to show that $$\mathrm { P } ( R = S ) = \frac { p } { 2 - p }$$
OCR S1 2011 January Q2
11 marks Moderate -0.8
2 The random variable \(X\) has the distribution \(\operatorname { Geo } ( 0.2 )\). Find
  1. \(\mathrm { P } ( X = 3 )\),
  2. \(\mathrm { P } ( 3 \leqslant X \leqslant 5 )\),
  3. \(\mathrm { P } ( X > 4 )\). Two independent values of \(X\) are found.
  4. Find the probability that the total of these two values is 3 .
OCR S1 2011 January Q3
12 marks Moderate -0.8
3 A firm wishes to assess whether there is a linear relationship between the annual amount spent on advertising, \(\pounds x\) thousand, and the annual profit, \(\pounds y\) thousand. A summary of the figures for 12 years is as follows. $$n = 12 \quad \Sigma x = 86.6 \quad \Sigma y = 943.8 \quad \Sigma x ^ { 2 } = 658.76 \quad \Sigma y ^ { 2 } = 83663.00 \quad \Sigma x y = 7351.12$$
  1. Calculate the product moment correlation coefficient, showing that it is greater than 0.9 .
  2. Comment briefly on this value in this context.
  3. A manager claims that this result shows that spending more money on advertising in the future will result in greater profits. Make two criticisms of this claim.
  4. Calculate the equation of the regression line of \(y\) on \(x\).
  5. Estimate the annual profit during a year when \(\pounds 7400\) was spent on advertising.
OCR S1 2011 January Q4
7 marks Moderate -0.8
4 Jenny and Omar are each allowed two attempts at a high jump.
  1. The probability that Jenny will succeed on her first attempt is 0.6 . If she fails on her first attempt, the probability that she will succeed on her second attempt is 0.7 . Calculate the probability that Jenny will succeed.
  2. The probability that Omar will succeed on his first attempt is \(p\). If he fails on his first attempt, the probability that he will succeed on his second attempt is also \(p\). The probability that he succeeds is 0.51 . Find \(p\). \(530 \%\) of packets of Natural Crunch Crisps contain a free gift. Jan buys 5 packets each week.
OCR S1 2011 January Q6
10 marks Moderate -0.8
6
  1. The diagram shows 7 cards, each with a digit printed on it. The digits form a 7 -digit number.
    1333559
    How many different 7 -digit numbers can be formed using these cards?
  2. The diagram below shows 5 white cards and 10 grey cards, each with a letter printed on it. \includegraphics[max width=\textwidth, alt={}, center]{98ac515d-fd47-4864-afd6-321e9848d6cb-04_398_801_596_632} From these cards, 3 white cards and 4 grey cards are selected at random without regard to order.
    (a) How many selections of seven cards are possible?
    (b) Find the probability that the seven cards include exactly one card showing the letter A .
OCR S1 2011 January Q7
5 marks Easy -1.2
7 The probability distribution of a discrete random variable, \(X\), is shown below.
\(x\)02
\(\mathrm { P } ( X = x )\)\(a\)\(1 - a\)
  1. Find \(\mathrm { E } ( X )\) in terms of \(a\).
  2. Show that \(\operatorname { Var } ( X ) = 4 a ( 1 - a )\).
OCR S1 2012 January Q1
4 marks Easy -1.3
1 The probability distribution of a random variable \(X\) is shown in the table.
\(x\)1234
\(\mathrm { P } ( X = x )\)0.10.3\(2 p\)\(p\)
  1. Find \(p\).
  2. Find \(\mathrm { E } ( X )\).
OCR S1 2012 January Q2
10 marks Easy -1.8
2 In an experiment, the percentage sand content, \(y\), of soil in a given region was measured at nine different depths, \(x \mathrm {~cm}\), taken at intervals of 6 cm from 0 cm to 48 cm . The results are summarised below. $$n = 9 \quad \Sigma x = 216 \quad \Sigma x ^ { 2 } = 7344 \quad \Sigma y = 512.4 \quad \Sigma y ^ { 2 } = 30595 \quad \Sigma x y = 10674$$
  1. State, with a reason, which variable is the independent variable.
  2. Calculate the product moment correlation coefficient between \(x\) and \(y\).
  3. (a) Calculate the equation of the appropriate regression line.
    (b) This regression line is used to estimate the percentage sand content at depths of 25 cm and 100 cm . Comment on the reliability of each of these estimates. You are not asked to find the estimates.
OCR S1 2012 January Q3
6 marks Standard +0.3
3 A random variable \(X\) has the distribution \(\mathrm { B } ( 13,0.12 )\).
  1. Find \(\mathrm { P } ( X < 2 )\). Two independent values of \(X\) are found.
  2. Find the probability that exactly one of these values is equal to 2 .
OCR S1 2012 January Q4
8 marks Standard +0.8
4
  1. The table gives the heights and masses of 5 people.
    Person\(A\)\(B\)\(C\)\(D\)\(E\)
    Height (m)1.721.631.771.681.74
    Mass (kg)7562646070
    Calculate Spearman's rank correlation coefficient.
  2. In an art competition the value of Spearman's rank correlation coefficient, \(r _ { s }\), calculated from two judges’ rankings was 0.75 . A late entry for the competition was received and both judges ranked this entry lower than all the others. By considering the formula for \(r _ { s }\), explain whether the new value of \(r _ { s }\) will be less than 0.75 , equal to 0.75 , or greater than 0.75 .
OCR S1 2012 January Q5
11 marks Moderate -0.8
5 At a certain resort the number of hours of sunshine, measured to the nearest hour, was recorded on each of 21 days. The results are summarised in the table.
Hours of sunshine0\(1 - 3\)\(4 - 6\)\(7 - 9\)\(10 - 15\)
Number of days06942
The diagram shows part of a histogram to illustrate the data. The scale on the frequency density axis is 2 cm to 1 unit. \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-3_944_1778_699_148}
  1. (a) Calculate the frequency density of the \(1 - 3\) class.
    (b) Fred wishes to draw the block for the 10 - 15 class on the same diagram. Calculate the height, in centimetres, of this block.
  2. A cumulative frequency graph is to be drawn. Write down the coordinates of the first two points that should be plotted. You are not asked to draw the graph.
  3. (a) Calculate estimates of the mean and standard deviation of the number of hours of sunshine.
    (b) Explain why your answers are only estimates.
OCR S1 2012 January Q6
5 marks Moderate -0.8
6 The diagrams illustrate all or part of the probability distributions of the discrete random variables \(V , W , X , Y\) and \(Z\). \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_419_365_370_296} \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_419_376_370_838} \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_419_362_370_1400} \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_421_359_879_580} \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_419_355_881_1142}
  1. One of these variables has the distribution \(\operatorname { Geo } \left( \frac { 1 } { 2 } \right)\). State, with a reason, which variable this is.
  2. One of these variables has the distribution \(\mathrm { B } \left( 4 , \frac { 1 } { 2 } \right)\). State, with reasons, which variable this is. \(760 \%\) of the voters at a certain polling station are women. Voters enter the polling station one at a time. The number of voters who enter, up to and including the first woman, is denoted by \(X\).
OCR S1 2012 January Q8
8 marks Moderate -0.8
8 On average, half the plants of a particular variety produce red flowers and the rest produce blue flowers.
  1. Ann chooses 8 plants of this variety at random. Find the probability that more than 6 plants produce red flowers.
  2. Karim chooses 22 plants of this variety at random.
    (a) Find the probability that the number of these plants that produce blue flowers is equal to the number that produce red flowers.
    (b) Hence find the probability that the number of these plants that produce blue flowers is greater than the number that produce red flowers.
OCR S1 2012 January Q9
12 marks Moderate -0.3
9 A bag contains 9 discs numbered 1, 2, 3, 4, 5, 6, 7, 8, 9 .
  1. Andrea chooses 4 discs at random, without replacement, and places them in a row.
    (a) How many different 4 -digit numbers can be made?
    (b) How many different odd 4-digit numbers can be made?
  2. Andrea's 4 discs are put back in the bag. Martin then chooses 4 discs at random, without replacement. Find the probability that
    (a) the 4 digits include at least 3 odd digits,
    (b) the 4 digits add up to 28 .
OCR S1 2011 June Q1
7 marks Moderate -0.8
1 Five salesmen from a certain firm were selected at random for a survey. For each salesman, the annual income, \(x\) thousand pounds, and the distance driven last year, \(y\) thousand miles, were recorded. The results were summarised as follows. $$n = 5 \quad \Sigma x = 251 \quad \Sigma x ^ { 2 } = 14323 \quad \Sigma y = 65 \quad \Sigma y ^ { 2 } = 855 \quad \Sigma x y = 3247$$
  1. (a) Show that the product moment correlation coefficient, \(r\), between \(x\) and \(y\) is - 0.122 , correct to 3 significant figures.
    (b) State what this value of \(r\) shows about the relationship between annual income and distance driven last year for these five salesmen.
    (c) It was decided to recalculate \(r\) with the distances measured in kilometres instead of miles. State what effect, if any, this would have on the value of \(r\).
  2. Another salesman from the firm is selected at random. His annual income is known to be \(\pounds 52000\), but the distance that he drove last year is unknown. In order to estimate this distance, a regression line based on the above data is used. Comment on the reliability of such an estimate.
OCR S1 2011 June Q2
5 marks Easy -1.2
2 The orders in which 4 contestants, \(P , Q , R\) and \(S\), were placed in two competitions are shown in the table.
Position1st2nd3rd4th
Competition 1\(Q\)\(R\)\(S\)\(P\)
Competition 2\(Q\)\(P\)\(R\)\(S\)
Calculate Spearman's rank correlation coefficient between these two orders.