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OCR MEI S1 2010 January Q5
3 marks Easy -1.2
My credit card has a 4-digit code called a PIN. You should assume that any 4-digit number from 0000 to 9999 can be a PIN.
  1. If I cannot remember any digits and guess my number, find the probability that I guess it correctly. [1]
In fact my PIN consists of four different digits. I can remember all four digits, but cannot remember the correct order.
  1. If I now guess my number, find the probability that I guess it correctly. [2]
OCR MEI S1 2010 January Q6
4 marks Easy -1.2
Three prizes, one for English, one for French and one for Spanish, are to be awarded in a class of 20 students. Find the number of different ways in which the three prizes can be awarded if
  1. no student may win more than 1 prize, [2]
  2. no student may win all 3 prizes. [2]
OCR MEI S1 2010 January Q7
19 marks Moderate -0.8
A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm, of these pears. \includegraphics{figure_7}
  1. Calculate the number of pears which are between 90 and 100 mm long. [2]
  2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate. [4]
  3. Calculate an estimate of the standard deviation. [3]
  4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers. [4]
  5. Name the type of skewness of the distribution. [1]
  6. Illustrate the data using a cumulative frequency diagram. [5]
OCR MEI S1 2010 January Q8
17 marks Standard +0.3
An environmental health officer monitors the air pollution level in a city street. Each day the level of pollution is classified as low, medium or high. The probabilities of each level of pollution on a randomly chosen day are as given in the table.
Pollution levelLowMediumHigh
Probability0.50.350.15
  1. Three days are chosen at random. Find the probability that the pollution level is
    1. low on all 3 days, [2]
    2. low on at least one day, [2]
    3. low on one day, medium on another day, and high on the other day. [3]
  2. Ten days are chosen at random. Find the probability that
    1. there are no days when the pollution level is high, [2]
    2. there is exactly one day when the pollution level is high. [3]
The environmental health officer believes that pollution levels will be low more frequently in a different street. On 20 randomly selected days she monitors the pollution level in this street and finds that it is low on 15 occasions.
  1. Carry out a test at the 5% level to determine if there is evidence to suggest that she is correct. Use hypotheses \(H_0: p = 0.5\), \(H_1: p > 0.5\), where \(p\) represents the probability that the pollution level in this street is low. Explain why \(H_1\) has this form. [5]
OCR MEI S1 2011 January Q1
3 marks Easy -1.8
The stem and leaf diagram shows the weights, rounded to the nearest 10 grams, of 25 female iguanas. \begin{align} 8 &| 3 \quad 9
9 &| 3 \quad 5 \quad 6 \quad 6 \quad 6 \quad 8 \quad 9 \quad 9
10 &| 0 \quad 2 \quad 2 \quad 3 \quad 4 \quad 6 \quad 9
11 &| 2 \quad 4 \quad 7 \quad 8
12 &| 3 \quad 4 \quad 5
13 &| 2 \end{align} Key: \(11|2\) represents 1120 grams
  1. Find the mode and the median of the data. [2]
  2. Identify the type of skewness of the distribution. [1]
OCR MEI S1 2011 January Q2
4 marks Moderate -0.8
The table shows all the possible products of the scores on two fair four-sided dice.
Score on second die
1234
\multirow{4}{*}{\rotatebox{90}{Score on first die}} 11234
\cline{2-5} 22468
\cline{2-5} 336912
\cline{2-5} 4481216
  1. Find the probability that the product of the two scores is less than 10. [1]
  2. Show that the events 'the score on the first die is even' and 'the product of the scores on the two dice is less than 10' are not independent. [3]
OCR MEI S1 2011 January Q3
6 marks Moderate -0.8
There are 13 men and 10 women in a running club. A committee of 3 men and 3 women is to be selected.
  1. In how many different ways can the three men be selected? [2]
  2. In how many different ways can the whole committee be selected? [2]
  3. A random sample of 6 people is selected from the running club. Find the probability that this sample consists of 3 men and 3 women. [2]
OCR MEI S1 2011 January Q4
7 marks Standard +0.3
The probability distribution of the random variable \(X\) is given by the formula $$\text{P}(X = r) = kr(r + 1) \quad \text{for } r = 1, 2, 3, 4, 5.$$
  1. Show that \(k = \frac{1}{70}\). [2]
  2. Find E\((X)\) and Var\((X)\). [5]
OCR MEI S1 2011 January Q5
8 marks Moderate -0.8
Andy can walk to work, travel by bike or travel by bus. The tree diagram shows the probabilities of any day being dry or wet and the corresponding probabilities for each of Andy's methods of travel. \includegraphics{figure_5} A day is selected at random. Find the probability that
  1. the weather is wet and Andy travels by bus, [2]
  2. Andy walks or travels by bike, [3]
  3. the weather is dry given that Andy walks or travels by bike. [3]
OCR MEI S1 2011 January Q6
8 marks Moderate -0.8
A survey is being carried out into the carbon footprint of individual citizens. As part of the survey, 100 citizens are asked whether they have attempted to reduce their carbon footprint by any of the following methods.
  • Reducing car use
  • Insulating their homes
  • Avoiding air travel
The numbers of citizens who have used each of these methods are shown in the Venn diagram. \includegraphics{figure_6} One of the citizens is selected at random.
  1. Find the probability that this citizen
    1. has avoided air travel, [1]
    2. has used at least two of the three methods. [2]
  2. Given that the citizen has avoided air travel, find the probability that this citizen has reduced car use. [2]
Three of the citizens are selected at random.
  1. Find the probability that none of them have avoided air travel. [3]
OCR MEI S1 2011 January Q7
19 marks Moderate -0.3
The incomes of a sample of 918 households on an island are given in the table below.
Income (x thousand pounds)\(0 \leqslant x \leqslant 20\)\(20 < x \leqslant 40\)\(40 < x \leqslant 60\)\(60 < x \leqslant 100\)\(100 < x \leqslant 200\)
Frequency23836514212845
  1. Draw a histogram to illustrate the data. [5]
  2. Calculate an estimate of the mean income. [3]
  3. Calculate an estimate of the standard deviation of the incomes. [4]
  4. Use your answers to parts (ii) and (iii) to show there are almost certainly some outliers in the sample. Explain whether or not it would be appropriate to exclude the outliers from the calculation of the mean and the standard deviation. [4]
  5. The incomes were converted into another currency using the formula \(y = 1.15x\). Calculate estimates of the mean and variance of the incomes in the new currency. [3]
OCR MEI S1 2011 January Q8
17 marks Standard +0.3
Mark is playing solitaire on his computer. The probability that he wins a game is 0.2, independently of all other games that he plays.
  1. Find the expected number of wins in 12 games. [2]
  2. Find the probability that
    1. he wins exactly 2 out of the next 12 games that he plays, [3]
    2. he wins at least 2 out of the next 12 games that he plays. [3]
  3. Mark's friend Ali also plays solitaire. Ali claims that he is better at winning games than Mark. In a random sample of 20 games played by Ali, he wins 7 of them. Write down suitable hypotheses for a test at the 5\% level to investigate whether Ali is correct. Give a reason for your choice of alternative hypothesis. Carry out the test. [9]
OCR MEI S1 2011 June Q1
5 marks Easy -1.3
In the Paris-Roubaix cycling race, there are a number of sections of cobbled road. The lengths of these sections, measured in metres, are illustrated in the histogram. \includegraphics{figure_1}
  1. Find the number of sections which are between 1000 and 2000 metres in length. [2]
  2. Name the type of skewness suggested by the histogram. [1]
  3. State the minimum and maximum possible values of the midrange. [2]
OCR MEI S1 2011 June Q2
5 marks Easy -1.3
I have 5 books, each by a different author. The authors are Austen, Brontë, Clarke, Dickens and Eliot.
  1. If I arrange the books in a random order on my bookshelf, find the probability that the authors are in alphabetical order with Austen on the left. [2]
  2. If I choose two of the books at random, find the probability that I choose the books written by Austen and Brontë. [3]
OCR MEI S1 2011 June Q3
4 marks Easy -1.2
25% of the plants of a particular species have red flowers. A random sample of 6 plants is selected.
  1. Find the probability that there are no plants with red flowers in the sample. [2]
  2. If 50 random samples of 6 plants are selected, find the expected number of samples in which there are no plants with red flowers. [2]
OCR MEI S1 2011 June Q4
7 marks Moderate -0.8
Two fair six-sided dice are thrown. The random variable \(X\) denotes the difference between the scores on the two dice. The table shows the probability distribution of \(X\).
\(r\)012345
P(X = r)\(\frac{1}{6}\)\(\frac{5}{18}\)\(\frac{2}{9}\)\(\frac{1}{6}\)\(\frac{1}{9}\)\(\frac{1}{18}\)
  1. Draw a vertical line chart to illustrate the probability distribution. [2]
  2. Use a probability argument to show that
    1. P(X = 1) = \(\frac{5}{18}\). [2]
    2. P(X = 0) = \(\frac{1}{6}\). [1]
  3. Find the mean value of \(X\). [2]
OCR MEI S1 2011 June Q5
8 marks Moderate -0.8
In a recent survey, a large number of working people were asked whether they worked full-time or part-time, with part-time being defined as less than 25 hours per week. One of the respondents is selected at random. • \(W\) is the event that this person works part-time. • \(F\) is the event that this person is female. You are given that P(\(W\)) = 0.14, P(\(F\)) = 0.41 and P(\(W \cap F\)) = 0.11.
  1. Draw a Venn diagram showing the events \(W\) and \(F\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
  2. Determine whether the events \(W\) and \(F\) are independent. [2]
  3. Find P(\(W\) | \(F\)) and explain what this probability represents. [3]
OCR MEI S1 2011 June Q6
7 marks Moderate -0.8
The numbers of eggs laid by a sample of 70 female herring gulls are shown in the table.
Number of eggs1234
Frequency1040155
  1. Find the mean and standard deviation of the number of eggs laid per gull. [5]
  2. The sample did not include female herring gulls that laid no eggs. How would the mean and standard deviation change if these gulls were included? [2]
OCR MEI S1 2011 June Q7
18 marks Standard +0.3
Any patient who fails to turn up for an outpatient appointment at a hospital is described as a 'no-show'. At a particular hospital, on average 15% of patients are no-shows. A random sample of 20 patients who have outpatient appointments is selected.
  1. Find the probability that
    1. there is exactly 1 no-show in the sample, [3]
    2. there are at least 2 no-shows in the sample. [2]
The hospital management introduces a policy of telephoning patients before appointments. It is hoped that this will reduce the proportion of no-shows. In order to check this, a random sample of \(n\) patients is selected. The number of no-shows in the sample is recorded and a hypothesis test is carried out at the 5% level.
  1. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis. [4]
  2. In the case that \(n = 20\) and the number of no-shows in the sample is 1, carry out the test. [4]
  3. In another case, where \(n\) is large, the number of no-shows in the sample is 6 and the critical value for the test is 8. Complete the test. [3]
  4. In the case that \(n \leqslant 18\), explain why there is no point in carrying out the test at the 5% level. [2]
OCR MEI S1 2011 June Q8
18 marks Moderate -0.3
The heating quality of the coal in a sample of 50 sacks is measured in suitable units. The data are summarised below.
Heating quality (\(x\))9.1 \(\leqslant x <\) 9.39.3 \(< x \leqslant\) 9.59.5 \(< x \leqslant\) 9.79.7 \(< x \leqslant\) 9.99.9 \(< x \leqslant\) 10.1
Frequency5715167
  1. Draw a cumulative frequency diagram to illustrate these data. [5]
  2. Use the diagram to estimate the median and interquartile range of the data. [3]
  3. Show that there are no outliers in the sample. [3]
  4. Three of these 50 sacks are selected at random. Find the probability that
    1. in all three, the heating quality \(x\) is more than 9.5, [3]
    2. in at least two, the heating quality \(x\) is more than 9.5. [4]
OCR MEI S1 2014 June Q1
8 marks Easy -1.3
The ages, \(x\) years, of the senior members of a running club are summarised in the table below.
Age (\(x\))\(20 \leqslant x < 30\)\(30 \leqslant x < 40\)\(40 \leqslant x < 50\)\(50 \leqslant x < 60\)\(60 \leqslant x < 70\)\(70 \leqslant x < 80\)\(80 \leqslant x < 90\)
Frequency10304223951
  1. Draw a cumulative frequency diagram to illustrate the data. [5]
  2. Use your diagram to estimate the median and interquartile range of the data. [3]
OCR MEI S1 2014 June Q2
8 marks Moderate -0.8
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. [3]
  2. Find the probability that a randomly selected candidate is accepted. [2]
  3. Find the probability that a randomly selected candidate is retested at least once, given that this candidate is accepted. [3]
OCR MEI S1 2014 June Q3
6 marks Easy -1.2
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. [1]
  2. Find \(\mathrm{P}(L \cap R)\). [2]
  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. [3]
OCR MEI S1 2014 June Q4
6 marks Moderate -0.8
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. [3]
  2. Find the probability that the team will contain at least one girl and at least one boy. [3]
OCR MEI S1 2014 June Q5
8 marks Moderate -0.8
The probability distribution of the random variable \(X\) is given by the formula $$\mathrm{P}(X = r) = k + 0.01r^2 \text{ for } r = 1, 2, 3, 4, 5.$$
  1. Show that \(k = 0.09\). Using this value of \(k\), display the probability distribution of \(X\) in a table. [3]
  2. Find \(\mathrm{E}(X)\) and \(\mathrm{Var}(X)\). [5]