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CAIE S1 2011 November Q6
9 marks Moderate -0.8
6 There are a large number of students in Luttley College. \(60 \%\) of the students are boys. Students can choose exactly one of Games, Drama or Music on Friday afternoons. It is found that \(75 \%\) of the boys choose Games, \(10 \%\) of the boys choose Drama and the remainder of the boys choose Music. Of the girls, \(30 \%\) choose Games, \(55 \%\) choose Drama and the remainder choose Music.
  1. 6 boys are chosen at random. Find the probability that fewer than 3 of them choose Music.
  2. 5 Drama students are chosen at random. Find the probability that at least 1 of them is a boy.
  1. In a certain country, the daily minimum temperature, in \({ } ^ { \circ } \mathrm { C }\), in winter has the distribution \(\mathrm { N } ( 8,24 )\). Find the probability that a randomly chosen winter day in this country has a minimum temperature between \(7 ^ { \circ } \mathrm { C }\) and \(12 ^ { \circ } \mathrm { C }\). The daily minimum temperature, in \({ } ^ { \circ } \mathrm { C }\), in another country in winter has a normal distribution with mean \(\mu\) and standard deviation \(2 \mu\).
  2. Find the proportion of winter days on which the minimum temperature is below zero.
  3. 70 winter days are chosen at random. Find how many of these would be expected to have a minimum temperature which is more than three times the mean.
  4. The probability of the minimum temperature being above \(6 ^ { \circ } \mathrm { C }\) on any winter day is 0.0735 . Find the value of \(\mu\).
CAIE S1 2011 November Q1
6 marks Standard +0.3
1 The random variable \(X\) is normally distributed and is such that the mean \(\mu\) is three times the standard deviation \(\sigma\). It is given that \(\mathrm { P } ( X < 25 ) = 0.648\).
  1. Find the values of \(\mu\) and \(\sigma\).
  2. Find the probability that, from 6 random values of \(X\), exactly 4 are greater than 25 .
CAIE S1 2011 November Q2
6 marks Moderate -0.8
2 In a group of 30 teenagers, 13 of the 18 males watch 'Kops are Kids' on television and 3 of the 12 females watch 'Kops are Kids'.
  1. Find the probability that a person chosen at random from the group is either female or watches 'Kops are Kids' or both.
  2. Showing your working, determine whether the events 'the person chosen is male' and 'the person chosen watches Kops are Kids' are independent or not.
CAIE S1 2011 November Q3
9 marks Standard +0.3
3 A factory makes a large number of ropes with lengths either 3 m or 5 m . There are four times as many ropes of length 3 m as there are ropes of length 5 m .
  1. One rope is chosen at random. Find the expectation and variance of its length.
  2. Two ropes are chosen at random. Find the probability that they have different lengths.
  3. Three ropes are chosen at random. Find the probability that their total length is 11 m .
CAIE S1 2011 November Q4
9 marks Moderate -0.8
4 Mary saves her digital images on her computer in three separate folders named 'Family', 'Holiday' and 'Friends'. Her family folder contains 3 images, her holiday folder contains 4 images and her friends folder contains 8 images. All the images are different.
  1. Find in how many ways she can arrange these 15 images in a row across her computer screen if she keeps the images from each folder together.
  2. Find the number of different ways in which Mary can choose 6 of these images if there are 2 from each folder.
  3. Find the number of different ways in which Mary can choose 6 of these images if there are at least 3 images from the friends folder and at least 1 image from each of the other two folders.
CAIE S1 2011 November Q5
9 marks Moderate -0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{b72ace6b-d3d4-401d-bffe-403c9127f2a8-3_1157_1001_258_573} The cumulative frequency graph shows the annual salaries, in thousands of euros, of a random sample of 500 adults with jobs, in France. It has been plotted using grouped data. You may assume that the lowest salary is 5000 euros and the highest salary is 80000 euros.
  1. On graph paper, draw a box-and-whisker plot to illustrate these salaries.
  2. Comment on the salaries of the people in this sample.
  3. An 'outlier' is defined as any data value which is more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile.
    1. How high must a salary be in order to be classified as an outlier?
    2. Show that none of the salaries is low enough to be classified as an outlier.
CAIE S1 2011 November Q6
11 marks Standard +0.3
6 Human blood groups are identified by two parts. The first part is \(\mathrm { A } , \mathrm { B } , \mathrm { AB }\) or O and the second part (the Rhesus part) is + or - . In the UK, \(35 \%\) of the population are group \(\mathrm { A } + , 8 \%\) are \(\mathrm { B } + , 3 \%\) are \(\mathrm { AB } +\), \(37 \%\) are \(\mathrm { O } + , 7 \%\) are \(\mathrm { A } - , 2 \%\) are \(\mathrm { B } - , 1 \%\) are \(\mathrm { AB } -\) and \(7 \%\) are \(\mathrm { O } -\).
  1. A random sample of 9 people in the UK who are Rhesus + is taken. Find the probability that fewer than 3 are group \(\mathrm { O } +\).
  2. A random sample of 150 people in the UK is taken. Find the probability that more than 60 people are group A+.
CAIE S1 2012 November Q1
4 marks Moderate -0.8
1 Ashok has 3 green pens and 7 red pens. His friend Rod takes 3 of these pens at random, without replacement. Draw up a probability distribution table for the number of green pens Rod takes.
CAIE S1 2012 November Q2
5 marks Moderate -0.8
2 The amounts of money, \(x\) dollars, that 24 people had in their pockets are summarised by \(\Sigma ( x - 36 ) = - 60\) and \(\Sigma ( x - 36 ) ^ { 2 } = 227.76\). Find \(\Sigma x\) and \(\Sigma x ^ { 2 }\).
CAIE S1 2012 November Q3
6 marks Standard +0.3
3 Lengths of rolls of parcel tape have a normal distribution with mean 75 m , and 15\% of the rolls have lengths less than 73 m .
  1. Find the standard deviation of the lengths. Alison buys 8 rolls of parcel tape.
  2. Find the probability that fewer than 3 of these rolls have lengths more than 77 m .
CAIE S1 2012 November Q4
7 marks Moderate -0.8
4 Prices in dollars of 11 caravans in a showroom are as follows. \(\begin{array} { l l l l l l l l l l l } 16800 & 18500 & 17700 & 14300 & 15500 & 15300 & 16100 & 16800 & 17300 & 15400 & 16400 \end{array}\)
  1. Represent these prices by a stem-and-leaf diagram.
  2. Write down the lower quartile of the prices of the caravans in the showroom.
  3. 3 different caravans in the showroom are chosen at random and their prices are noted. Find the probability that 2 of these prices are more than the median and 1 is less than the lower quartile.
CAIE S1 2012 November Q5
7 marks Standard +0.3
5 A company set up a display consisting of 20 fireworks. For each firework, the probability that it fails to work is 0.05 , independently of other fireworks.
  1. Find the probability that more than 1 firework fails to work. The 20 fireworks cost the company \(\\) 24\( each. 450 people pay the company \)\\( 10\) each to watch the display. If more than 1 firework fails to work they get their money back.
  2. Calculate the expected profit for the company.
CAIE S1 2012 November Q6
9 marks Standard +0.3
6 Ana meets her friends once every day. For each day the probability that she is early is 0.05 and the probability that she is late is 0.75 . Otherwise she is on time.
  1. Find the probability that she is on time on fewer than 20 of the next 96 days.
  2. If she is early there is a probability of 0.7 that she will eat a banana. If she is late she does not eat a banana. If she is on time there is a probability of 0.4 that she will eat a banana. Given that for one particular meeting with friends she does not eat a banana, find the probability that she is on time.
CAIE S1 2012 November Q7
12 marks Standard +0.3
7
  1. In a sweet shop 5 identical packets of toffees, 4 identical packets of fruit gums and 9 identical packets of chocolates are arranged in a line on a shelf. Find the number of different arrangements of the packets that are possible if the packets of chocolates are kept together.
  2. Jessica buys 8 different packets of biscuits. She then chooses 4 of these packets.
    1. How many different choices are possible if the order in which Jessica chooses the 4 packets is taken into account? The 8 packets include 1 packet of chocolate biscuits and 1 packet of custard creams.
    2. How many different choices are possible if the order in which Jessica chooses the 4 packets is taken into account and the packet of chocolate biscuits and the packet of custard creams are both chosen?
  3. 9 different fruit pies are to be divided between 3 people so that each person gets an odd number of pies. Find the number of ways this can be done.
OCR S1 2005 January Q1
4 marks Easy -1.3
1 The scatter diagrams below illustrate three sets of bivariate data, \(A , B\) and \(C\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0c0a4ca-da0a-4c74-b8b1-bac4fd3f2487-2_440_428_360_317} \captionsetup{labelformat=empty} \caption{Set \(A\)}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0c0a4ca-da0a-4c74-b8b1-bac4fd3f2487-2_440_426_360_858} \captionsetup{labelformat=empty} \caption{Set \(B\)}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0c0a4ca-da0a-4c74-b8b1-bac4fd3f2487-2_435_424_365_1402} \captionsetup{labelformat=empty} \caption{Set \(C\)}
\end{figure} State, with an explanation in each case, which of the three sets of data has
  1. the largest,
  2. the smallest,
    value of the product moment correlation coefficient.
OCR S1 2005 January Q2
6 marks Easy -1.8
2 The back-to-back stem-and-leaf diagram below shows the number of hours of television watched per week by each of 15 boys and 15 girls. $$\begin{aligned} & \text { Boys Girls } \\ & \left. \begin{array} { r r r r r r r r | r r r r r r r r r r r r r } & 677664 & 4 & 3 & 0 & 0 & 5 & 5 & 6 & 677888 \end{array} \right\} \end{aligned}$$ Key: 4 | 2 | 2 means a boy who watched 24 hours and a girl who watched 22 hours of television per week.
  1. Find the median and the quartiles of the results for the boys.
  2. Give a reason why the median might be preferred to the mean in using an average to compare the two data sets.
  3. State one advantage, and one disadvantage, of using stem-and-leaf diagrams rather than box-andwhisker plots to represent the data.
OCR S1 2005 January Q3
6 marks Moderate -0.8
3 Two commentators gave ratings out of 100 for seven sports personalities. The ratings are shown in the table below.
Personality\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)
Commentator I73767865868291
Commentator II77787980868995
  1. Calculate Spearman's rank correlation coefficient for these ratings.
  2. State what your answer tells you about the ratings given by the two commentators.
OCR S1 2005 January Q4
7 marks Easy -1.3
4 The table below shows the probability distribution of the random variable \(X\).
\(x\)- 2- 1012
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 5 }\)\(k\)\(\frac { 2 } { 5 }\)\(\frac { 1 } { 10 }\)
  1. Find the value of the constant \(k\).
  2. Calculate the values of \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR S1 2005 January Q5
8 marks Moderate -0.8
5 On average 1 in 20 members of the population of this country has a particular DNA feature. Members of the population are selected at random until one is found who has this feature.
  1. Find the probability that the first person to have this feature is
    1. the sixth person selected,
    2. not among the first 10 people selected.
    3. Find the expected number of people selected.
OCR S1 2005 January Q6
7 marks Moderate -0.3
6 Louise and Marie play a series of tennis matches. It is given that, in any match, the probability that Louise wins the first two sets is \(\frac { 3 } { 8 }\).
  1. Find the probability that, in 5 randomly chosen matches, Louise wins the first two sets in exactly 2 of the matches. It is also given that Louise and Marie are equally likely to win the first set.
  2. Show that P (Louise wins the second set, given that she won the first set) \(= \frac { 3 } { 4 }\).
  3. The probability that Marie wins the first two sets is \(\frac { 1 } { 3 }\). Find P(Marie wins the second set, given that she won the first set).
OCR S1 2005 January Q7
9 marks Easy -1.2
7 It is known that, on average, one match box in 10 contains fewer than 42 matches. Eight boxes are selected, and the number of boxes that contain fewer than 42 matches is denoted by \(Y\).
  1. State two conditions needed to model \(Y\) by a binomial distribution. Assume now that a binomial model is valid.
  2. Find
    1. \(\mathrm { P } ( Y = 0 )\),
    2. \(\mathrm { P } ( Y \geqslant 2 )\).
    3. On Wednesday 8 boxes are selected, and on Thursday another 8 boxes are selected. Find the probability that on one of these days the number of boxes containing fewer than 42 matches is 0 , and that on the other day the number is 2 or more.
OCR S1 2005 January Q8
10 marks Moderate -0.8
8 An examination paper consists of 8 questions, of which one is on geometric distributions and one is on binomial distributions.
  1. If the 8 questions are arranged in a random order, find the probability that the question on geometric distributions is next to the question on binomial distributions. Four of the questions, including the one on geometric distributions, are worth 7 marks each, and the remaining four questions, including the one on binomial distributions, are worth 9 marks each. The 7-mark questions are the first four questions on the paper, but are arranged in random order. The 9-mark questions are the last four questions, but are arranged in random order. Find the probability that
  2. the questions on geometric distributions and on binomial distributions are next to one another,
  3. the questions on geometric distributions and on binomial distributions are separated by at least 2 other questions.
OCR S1 2005 January Q9
15 marks Standard +0.3
9 Five observations of bivariate data produce the following results, denoted as ( \(x _ { i } , y _ { i }\) ) for \(i = 1,2,3,4,5\). $$\begin{aligned} & ( 13,2.7 ) \\ & { \left[ \Sigma x = 90 , \Sigma y = 15.0 , \Sigma x ^ { 2 } = 1720 , \Sigma y ^ { 2 } = 46.86 , \Sigma x y = 264.0 . \right] } \end{aligned}$$
  1. Show that the regression line of \(y\) on \(x\) has gradient - 0.06 , and find its equation in the form \(y = a + b x\).
  2. The regression line is used to estimate the value of \(y\) corresponding to \(x = 20\), but the value \(x = 20\) is accurate only to the nearest whole number. Calculate the difference between the largest and the smallest values that the estimated value of \(y\) could take. The numbers \(e _ { 1 } , e _ { 2 } , e _ { 3 } , e _ { 4 } , e _ { 5 }\) are defined by $$e _ { i } = a + b x _ { i } - y _ { i } \quad \text { for } i = 1,2,3,4,5$$
  3. The values of \(e _ { 1 } , e _ { 2 }\) and \(e _ { 3 }\) are \(0.6 , - 0.7\) and 0.2 respectively. Calculate the values of \(e _ { 4 }\) and \(e _ { 5 }\).
  4. Calculate the value of \(e _ { 1 } ^ { 2 } + e _ { 2 } ^ { 2 } + e _ { 3 } ^ { 2 } + e _ { 4 } ^ { 2 } + e _ { 5 } ^ { 2 }\) and explain the relevance of this quantity to the regression line found in part (i).
  5. Find the mean and the variance of \(e _ { 1 } , e _ { 2 } , e _ { 3 } , e _ { 4 } , e _ { 5 }\).
OCR S1 2007 January Q1
4 marks Easy -1.8
1 Part of the probability distribution of a variable, \(X\), is given in the table.
\(x\)0123
\(\mathrm { P } ( X = x )\)\(\frac { 3 } { 10 }\)\(\frac { 1 } { 5 }\)\(\frac { 2 } { 5 }\)
  1. Find \(\mathrm { P } ( X = 0 )\).
  2. Find \(\mathrm { E } ( X )\).
OCR S1 2007 January Q2
6 marks Moderate -0.8
2 The table contains data concerning five households selected at random from a certain town.
Number of people in the household23357
Number of cars belonging to people in the household11324
  1. Calculate the product moment correlation coefficient, \(r\), for the data in the table.
  2. Give a reason why it would not be sensible to use your answer to draw a conclusion about all the households in the town.