Questions S1 (1967 questions)

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OCR MEI S1 2014 June Q2
8 marks Standard +0.3
2 Candidates applying for jobs in a large company take an aptitude test, as a result of which they are either accepted, rejected or retested, with probabilities \(0.2,0.5\) and 0.3 respectively. When a candidate is retested for the first time, the three possible outcomes and their probabilities remain the same as for the original test. When a candidate is retested for the second time there are just two possible outcomes, accepted or rejected, with probabilities 0.4 and 0.6 respectively.
  1. Draw a probability tree diagram to illustrate the outcomes.
  2. Find the probability that a randomly selected candidate is accepted.
  3. Find the probability that a randomly selected candidate is retested at least once, given that this candidate is accepted.
OCR MEI S1 2014 June Q3
6 marks Moderate -0.8
3 Each weekday, Marta travels to school by bus. Sometimes she arrives late.
  • \(L\) is the event that Marta arrives late.
  • \(R\) is the event that it is raining.
You are given that \(\mathrm { P } ( L ) = 0.15 , \mathrm { P } ( R ) = 0.22\) and \(\mathrm { P } ( L \mid R ) = 0.45\).
  1. Use this information to show that the events \(L\) and \(R\) are not independent.
  2. Find \(\mathrm { P } ( L \cap R )\).
  3. Draw a Venn diagram showing the events \(L\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram.
OCR MEI S1 2014 June Q4
6 marks Moderate -0.3
4 There are 16 girls and 14 boys in a class. Four of them are to be selected to form a quiz team. The team is to be selected at random.
  1. Find the probability that all 4 members of the team will be girls.
  2. Find the probability that the team will contain at least one girl and at least one boy.
OCR MEI S1 2014 June Q5
8 marks Moderate -0.8
5 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k + 0.01 r ^ { 2 } \text { for } r = 1,2,3,4,5 \text {. }$$
  1. Show that \(k = 0.09\). Using this value of \(k\), display the probability distribution of \(X\) in a table.
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 2014 June Q7
19 marks Challenging +1.2
7 It is known that on average \(85 \%\) of seeds of a particular variety of tomato will germinate. Ramesh selects 15 of these seeds at random and sows them.
  1. (A) Find the probability that exactly 12 germinate.
    (B) Find the probability that fewer than 12 germinate. The following year Ramesh finds that he still has many seeds left. Because the seeds are now one year old, he suspects that the germination rate will be lower. He conducts a trial by randomly selecting \(n\) of these seeds and sowing them. He then carries out a hypothesis test at the \(1 \%\) significance level to investigate whether he is correct.
  2. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis.
  3. In a trial with \(n = 20\), Ramesh finds that 13 seeds germinate. Carry out the test.
  4. Suppose instead that Ramesh conducts the trial with \(n = 50\), and finds that 33 seeds germinate. Given that the critical value for the test in this case is 35 , complete the test.
  5. If \(n\) is small, there is no point in carrying out the test at the \(1 \%\) significance level, as the null hypothesis cannot be rejected however many seeds germinate. Find the least value of \(n\) for which the null hypothesis can be rejected, quoting appropriate probabilities to justify your answer.
OCR MEI S1 2015 June Q2
5 marks Easy -1.3
2 A survey is being carried out into the sports viewing habits of people in a particular area. As part of the survey, 250 people are asked which of the following sports they have watched on television in the past month.
  • Football
  • Cycling
  • Rugby
The numbers of people who have watched these sports are shown in the Venn diagram.
\includegraphics[max width=\textwidth, alt={}, center]{6015ae6c-bf76-4a0c-af0f-5c53f9c5ed2a-2_723_917_1183_575} One of the people is selected at random.
  1. Find the probability that this person has in the past month
    (A) watched cycling but not football,
    (B) watched either one or two of the three sports.
  2. Given that this person has watched cycling, find the probability that this person has not watched football.
OCR MEI S1 2015 June Q3
3 marks Easy -1.2
3 A normal pack of 52 playing cards contains 4 aces. A card is drawn at random from the pack. It is then replaced and the pack is shuffled, after which another card is drawn at random.
  1. Find the probability that neither card is an ace.
  2. This process is repeated 10 times. Find the expected number of times for which neither card is an ace.
OCR MEI S1 2015 June Q4
6 marks Moderate -0.8
4 A rugby team of 15 people is to be selected from a squad of 25 players.
  1. How many different teams are possible?
  2. In fact the team has to consist of 8 forwards and 7 backs. If 13 of the squad are forwards and the other 12 are backs, how many different teams are now possible?
  3. Find the probability that, if the team is selected at random from the squad of 25 players, it contains the correct numbers of forwards and backs.
OCR MEI S1 2015 June Q5
8 marks Easy -1.2
5 At a tourist information office the numbers of people seeking information each hour over the course of a 12-hour day are shown below. $$\begin{array} { l l l l l l l l l l l l } 6 & 25 & 38 & 39 & 31 & 18 & 35 & 31 & 33 & 15 & 21 & 28 \end{array}$$
  1. Construct a sorted stem and leaf diagram to represent these data.
  2. State the type of skewness suggested by your stem and leaf diagram.
  3. For these data find the median, the mean and the mode. Comment on the usefulness of the mode in this case.
OCR MEI S1 2015 June Q6
8 marks Standard +0.3
6 Three fair six-sided dice are thrown. The random variable \(X\) represents the highest of the three scores on the dice.
  1. Show that \(\mathrm { P } ( X = 6 ) = \frac { 91 } { 216 }\). The table shows the probability distribution of \(X\).
    \(r\)123456
    \(\mathrm { P } ( X = r )\)\(\frac { 1 } { 216 }\)\(\frac { 7 } { 216 }\)\(\frac { 19 } { 216 }\)\(\frac { 37 } { 216 }\)\(\frac { 61 } { 216 }\)\(\frac { 91 } { 216 }\)
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 2015 June Q7
17 marks Standard +0.3
7 A drug for treating a particular minor illness cures, on average, \(78 \%\) of patients. Twenty people with this minor illness are selected at random and treated with the drug.
  1. \(( A )\) Find the probability that exactly 19 patients are cured.
    (B) Find the probability that at most 18 patients are cured.
    \(( C )\) Find the expected number of patients who are cured.
  2. A pharmaceutical company is trialling a new drug to treat this illness. Researchers at the company hope that a higher percentage of patients will be cured when given this new drug. Twenty patients are selected at random, and given the new drug. Of these, 19 are cured. Carry out a hypothesis test at the \(1 \%\) significance level to investigate whether there is any evidence to suggest that the new drug is more effective than the old one.
  3. If the researchers had chosen to carry out the hypothesis test at the \(5 \%\) significance level, what would the result have been? Justify your answer.
OCR MEI S1 2015 June Q8
19 marks Standard +0.3
8 The box and whisker plot below summarises the weights in grams of the 20 chocolates in a box.
\includegraphics[max width=\textwidth, alt={}, center]{6015ae6c-bf76-4a0c-af0f-5c53f9c5ed2a-4_287_1177_319_427}
  1. Find the interquartile range of the data and hence determine whether there are any outliers at either end of the distribution. Ben buys a box of these chocolates each weekend. The chocolates all look the same on the outside, but 7 of them have orange centres, 6 have cherry centres, 4 have coffee centres and 3 have lemon centres. One weekend, each of Ben's 3 children eats one of the chocolates, chosen at random.
  2. Calculate the probabilities of the following events. A: all 3 chocolates have orange centres
    \(B\) : all 3 chocolates have the same centres
  3. Find \(\mathrm { P } ( A \mid B )\) and \(\mathrm { P } ( B \mid A )\). The following weekend, Ben buys an identical box of chocolates and again each of his 3 children eats one of the chocolates, chosen at random.
  4. Find the probability that, on both weekends, the 3 chocolates that they eat all have orange centres.
  5. Ben likes all of the chocolates except those with cherry centres. On another weekend he is the first of his family to eat some of the chocolates. Find the probability that he has to select more than 2 chocolates before he finds one that he likes. \section*{END OF QUESTION PAPER} \section*{OCR
    Oxford Cambridge and RSA}
OCR S1 2013 January Q1
7 marks Easy -1.2
1 When a four-sided spinner is spun, the number on which it lands is denoted by \(X\), where \(X\) is a random variable taking values \(2,4,6\) and 8 . The spinner is biased so that \(\mathrm { P } ( X = x ) = k x\), where \(k\) is a constant.
  1. Show that \(\mathrm { P } ( X = 6 ) = \frac { 3 } { 10 }\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  3. Kathryn is allowed three attempts at a high jump. If she succeeds on any attempt, she does not jump again. The probability that she succeeds on her first attempt is \(\frac { 3 } { 4 }\). If she fails on her first attempt, the probability that she succeeds on her second attempt is \(\frac { 3 } { 8 }\). If she fails on her first two attempts, the probability that she succeeds on her third attempt is \(\frac { 3 } { 16 }\). Find the probability that she succeeds.
  4. Khaled is allowed two attempts to pass an examination. If he succeeds on his first attempt, he does not make a second attempt. The probability that he passes at the first attempt is 0.4 and the probability that he passes on either the first or second attempt is 0.58 . Find the probability that he passes on the second attempt, given that he failed on the first attempt.
OCR S1 2013 January Q3
12 marks Moderate -0.3
3 The Gross Domestic Product per Capita (GDP), \(x\) dollars, and the Infant Mortality Rate per thousand (IMR), \(y\), of 6 African countries were recorded and summarised as follows. $$n = 6 \quad \sum x = 7000 \quad \sum x ^ { 2 } = 8700000 \quad \sum y = 456 \quad \sum y ^ { 2 } = 36262 \quad \sum x y = 509900$$
  1. Calculate the equation of the regression line of \(y\) on \(x\) for these 6 countries. The original data were plotted on a scatter diagram and the regression line of \(y\) on \(x\) was drawn, as shown below.
    \includegraphics[max width=\textwidth, alt={}, center]{13d8d940-fd63-4b62-bd7a-aa7174f6af4b-3_721_1246_680_408}
  2. The GDP for another country, Tanzania, is 1300 dollars. Use the regression line in the diagram to estimate the IMR of Tanzania.
  3. The GDP for Nigeria is 2400 dollars. Give two reasons why the regression line is unlikely to give a reliable estimate for the IMR for Nigeria.
  4. The actual value of the IMR for Tanzania is 96. The data for Tanzania \(( x = 1300 , y = 96 )\) is now included with the original 6 countries. Calculate the value of the product moment correlation coefficient, \(r\), for all 7 countries.
  5. The IMR is now redefined as the infant mortality rate per hundred instead of per thousand, and the value of \(r\) is recalculated for all 7 countries. Without calculation state what effect, if any, this would have on the value of \(r\) found in part (iv).
OCR S1 2013 January Q4
10 marks Moderate -0.8
4
  1. How many different 3-digit numbers can be formed using the digits 1, 2 and 3 when
    (a) no repetitions are allowed,
    (b) any repetitions are allowed,
    (c) each digit may be included at most twice?
  2. How many different 4-digit numbers can be formed using the digits 1, 2 and 3 when each digit may be included at most twice?
OCR S1 2013 January Q5
10 marks Standard +0.3
5 A random variable \(X\) has the distribution \(\mathrm { B } \left( 5 , \frac { 1 } { 4 } \right)\).
  1. Find
    (a) \(\mathrm { E } ( X )\),
    (b) \(\mathrm { P } ( X = 2 )\).
  2. Two values of \(X\) are chosen at random. Find the probability that their sum is less than 2 .
  3. 10 values of \(X\) are chosen at random. Use an appropriate formula to find the probability that exactly 3 of these values are 2 s .
OCR S1 2013 January Q6
7 marks Moderate -0.3
6 The masses, \(x\) grams, of 800 apples are summarised in the histogram.
\includegraphics[max width=\textwidth, alt={}, center]{13d8d940-fd63-4b62-bd7a-aa7174f6af4b-4_592_1363_957_351}
  1. On the frequency density axis, 1 cm represents \(a\) units. Find the value of \(a\).
  2. Find an estimate of the median mass of the apples.
  3. Two judges rank \(n\) competitors, where \(n\) is an even number. Judge 2 reverses each consecutive pair of ranks given by Judge 1, as shown.
    Competitor\(C _ { 1 }\)\(C _ { 2 }\)\(C _ { 3 }\)\(C _ { 4 }\)\(C _ { 5 }\)\(C _ { 6 }\)\(\ldots \ldots\)\(C _ { n - 1 }\)\(C _ { n }\)
    Judge 1 rank123456\(\ldots \ldots\)\(n - 1\)\(n\)
    Judge 2 rank214365\(\ldots \ldots\)\(n\)\(n - 1\)
    Given that the value of Spearman's coefficient of rank correlation is \(\frac { 63 } { 65 }\), find \(n\).
  4. An experiment produced some data from a bivariate distribution. The product moment correlation coefficient is denoted by \(r\), and Spearman's rank correlation coefficient is denoted by \(r _ { s }\).
    (a) Explain whether the statement $$r = 1 \Rightarrow r _ { s } = 1$$ is true or false.
    (b) Use a diagram to explain whether the statement $$r \neq 1 \Rightarrow r _ { s } \neq 1$$ is true or false. 8 Sandra makes repeated, independent attempts to hit a target. On each attempt, the probability that she succeeds is 0.1 .
  5. Find the probability that
    (a) the first time she succeeds is on her 5th attempt,
    (b) the first time she succeeds is after her 5th attempt,
    (c) the second time she succeeds is before her 4th attempt. Jill also makes repeated attempts to hit the target. Each attempt of either Jill or Sandra is independent. Each time that Jill attempts to hit the target, the probability that she succeeds is 0.2 . Sandra and Jill take turns attempting to hit the target, with Sandra going first.
  6. Find the probability that the first person to hit the target is Sandra, on her
    (a) 2nd attempt,
    (b) 10th attempt. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
OCR S1 2013 January Q8
13 marks Standard +0.8
8 Sandra makes repeated, independent attempts to hit a target. On each attempt, the probability that she succeeds is 0.1 .
  1. Find the probability that
    (a) the first time she succeeds is on her 5th attempt,
    (b) the first time she succeeds is after her 5th attempt,
    (c) the second time she succeeds is before her 4th attempt. Jill also makes repeated attempts to hit the target. Each attempt of either Jill or Sandra is independent. Each time that Jill attempts to hit the target, the probability that she succeeds is 0.2 . Sandra and Jill take turns attempting to hit the target, with Sandra going first.
  2. Find the probability that the first person to hit the target is Sandra, on her
    (a) 2nd attempt,
    (b) 10th attempt. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
    7
OCR S1 2013 June Q1
7 marks Easy -1.2
1 The lengths, in centimetres, of 18 snakes are given below. $$\begin{array} { l l l l l l l l l l l l l l l l l l } 24 & 62 & 20 & 65 & 27 & 67 & 69 & 32 & 40 & 53 & 55 & 47 & 33 & 45 & 55 & 56 & 49 & 58 \end{array}$$
  1. Draw an ordered stem-and-leaf diagram for the data.
  2. Find the mean and median of the lengths of the snakes.
  3. It was found that one of the lengths had been measured incorrectly. After this length was corrected, the median increased by 1 cm . Give two possibilities for the incorrect length and give a corrected value in each case.
OCR S1 2013 June Q2
7 marks Moderate -0.3
2
  1. The table shows the times, in minutes, spent by five students revising for a test, and the grades that they achieved in the test.
    StudentAnnBillCazDenEd
    Time revising0603510045
    GradeCDEBA
    Calculate Spearman's rank correlation coefficient.
  2. The table below shows the ranks given by two judges to four competitors.
    CompetitorPQRS
    Judge 1 rank1234
    Judge 2 rank3214
    Spearman's rank correlation coefficient for these ranks is denoted by \(r _ { s }\). With the same set of ranks for Judge 1, write down a different set of ranks for Judge 2 which gives the same value of \(r _ { s }\). There is no need to find the value of \(r _ { s }\).
OCR S1 2013 June Q3
10 marks Moderate -0.5
3 The probability distribution of a random variable \(X\) is shown.
\(x\)1357
\(\mathrm { P } ( X = x )\)0.40.30.20.1
  1. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  2. Three independent values of \(X\), denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\), are chosen. Given that \(X _ { 1 } + X _ { 2 } + X _ { 3 } = 19\), write down all the possible sets of values for \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) and hence find \(\mathrm { P } \left( X _ { 1 } = 7 \right)\).
  3. 11 independent values of \(X\) are chosen. Use an appropriate formula to find the probability that exactly 4 of these values are 5 s .
OCR S1 2013 June Q4
6 marks Moderate -0.8
4 At a stall in a fair, contestants have to estimate the mass of a cake. A group of 10 people made estimates, \(m \mathrm {~kg}\), and for each person the value of \(( m - 5 )\) was recorded. The mean and standard deviation of \(( m - 5 )\) were found to be 0.74 and 0.13 respectively.
  1. Write down the mean and standard deviation of \(m\). The mean and standard deviation of the estimates made by another group of 15 people were found to be 5.6 kg and 0.19 kg respectively.
  2. Calculate the mean of all 25 estimates.
  3. Fiona claims that if a group's estimates are more consistent, they are likely to be more accurate. Given that the true mass of the cake is 5.65 kg , comment on this claim.
OCR S1 2013 June Q5
9 marks Moderate -0.8
5 The table shows some of the values of the seasonally adjusted Unemployment Rate (UR), \(x \%\), and the Consumer Price Index (CPI), \(y \%\), in the United Kingdom from April 2008 to July 2010.
DateApril 2008July 2008October 2008January 2009April 2009July 2009October 2009January 2010April 2010July 2010
UR, \(x \%\)5.25.76.16.87.57.87.87.97.87.7
CPI, \(y \%\)3.04.44.53.02.31.81.53.53.73.1
These data are summarised below. $$n = 10 \quad \sum x = 70.3 \quad \sum x ^ { 2 } = 503.45 \quad \sum y = 30.8 \quad \sum y ^ { 2 } = 103.94 \quad \sum x y = 211.9$$
  1. Calculate the product moment correlation coefficient, \(r\), for the data, showing that \(- 0.6 < r < - 0.5\).
  2. Karen says "The negative value of \(r\) shows that when the Unemployment Rate increases, it causes the Consumer Price Index to decrease." Give a criticism of this statement.
  3. (a) Calculate the equation of the regression line of \(x\) on \(y\).
    (b) Use your equation to estimate the value of the Unemployment Rate in a month when the Consumer Price Index is 4.0\%.
OCR S1 2013 June Q6
7 marks Easy -1.2
6 The diagram shows five cards, each with a letter on it.
\includegraphics[max width=\textwidth, alt={}, center]{d06430a6-7957-4313-beea-bb320fadb282-4_113_743_315_662} The letters A and E are vowels; the letters B, C and D are consonants.
  1. Two of the five cards are chosen at random, without replacement. Find the probability that they both have vowels on them.
  2. The two cards are replaced. Now three of the five cards are chosen at random, without replacement. Find the probability that they include exactly one card with a vowel on it.
  3. The three cards are replaced. Now four of the five cards are chosen at random without replacement. Find the probability that they include the card with the letter B on it.
OCR S1 2013 June Q7
11 marks Standard +0.3
7 In a factory, an inspector checks a random sample of 30 mugs from a large batch and notes the number, \(X\), which are defective. He then deals with the batch as follows.
  • If \(X < 2\), the batch is accepted.
  • If \(X > 2\), the batch is rejected.
  • If \(X = 2\), the inspector selects another random sample of only 15 mugs from the batch. If this second sample contains 1 or more defective mugs, the batch is rejected. Otherwise the batch is accepted.
It is given that \(5 \%\) of mugs are defective.
  1. (a) Find the probability that the batch is rejected after just the first sample is checked.
    (b) Show that the probability that the batch is rejected is 0.327 , correct to 3 significant figures.
  2. Batches are checked one after another. Find the probability that the first batch to be rejected is either the 4th or the 5th batch that is checked.
  3. A bag contains 12 black discs, 10 white discs and 5 green discs. Three discs are drawn at random from the bag, without replacement. Find the probability that all three discs are of different colours.
  4. A bag contains 30 red discs and 20 blue discs. A second bag contains 50 discs, each of which is either red or blue. A disc is drawn at random from each bag. The probability that these two discs are of different colours is 0.54 . Find the number of red discs that were in the second bag at the start.