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OCR MEI S1 Q5
6 marks Moderate -0.8
The number, \(X\), of children per family in a certain city is modelled by the probability distribution P(\(X = r\)) = \(k(6 - r)(1 + r)\) for \(r = 0, 1, 2, 3, 4\).
  1. Copy and complete the following table and hence show that the value of \(k\) is \(\frac{1}{50}\). [3]
    \(r\)01234
    P(\(X = r\))\(6k\)\(10k\)
  2. Calculate E(\(X\)). [2]
  3. Hence write down the probability that a randomly selected family in this city has more than the mean number of children. [1]
OCR MEI S1 Q6
17 marks Moderate -0.8
The weights, \(w\) grams, of a random sample of 60 carrots of variety A are summarised in the table below.
Weight\(30 \leqslant w < 50\)\(50 \leqslant w < 60\)\(60 \leqslant w < 70\)\(70 \leqslant w < 80\)\(80 \leqslant w < 90\)
Frequency111018147
  1. Draw a histogram to illustrate these data. [5]
  2. Calculate estimates of the mean and standard deviation of \(w\). [4]
  3. Use your answers to part (ii) to investigate whether there are any outliers. [3]
The weights, \(x\) grams, of a random sample of 50 carrots of variety B are summarised as follows. \(n = 50\) \quad \(\sum x = 3624.5\) \quad \(\sum x^2 = 265416\)
  1. Calculate the mean and standard deviation of \(x\). [3]
  2. Compare the central tendency and variation of the weights of varieties A and B. [2]
OCR MEI S1 Q7
7 marks Moderate -0.8
A supermarket chain buys a batch of 10000 scratchcard draw tickets for sale in its stores. 50 of these tickets have a £10 prize, 20 of them have a £100 prize, one of them has a £5000 prize and all of the rest have no prize. This information is summarised in the frequency table below.
Prize money£0£10£100£5000
Frequency992950201
  1. Find the mean and standard deviation of the prize money per ticket. [4]
  2. I buy two of these tickets at random. Find the probability that I win either two £10 prizes or two £100 prizes. [3]
OCR MEI S1 Q1
8 marks Moderate -0.8
It is known that 8% of the population of a large city use a particular web browser. A researcher wishes to interview some people from the city who use this browser. He selects people at random, one at a time.
  1. Find the probability that the first person that he finds who uses this browser is
    1. the third person selected, [3]
    2. the second or third person selected. [2]
  2. Find the probability that at least one of the first 20 people selected uses this browser. [3]
OCR MEI S1 Q2
8 marks Standard +0.3
Jimmy and Alan are playing a tennis match against each other. The winner of the match is the first player to win three sets. Jimmy won the first set and Alan won the second set. For each of the remaining sets, the probability that Jimmy wins a set is • 0.7 if he won the previous set, • 0.4 if Alan won the previous set. It is not possible to draw a set.
  1. Draw a probability tree diagram to illustrate the possible outcomes for each of the remaining sets. [3]
  2. Find the probability that Alan wins the match. [3]
  3. Find the probability that the match ends after exactly four sets have been played. [2]
OCR MEI S1 Q3
6 marks Moderate -0.8
In a food survey, a large number of people are asked whether they like tomato soup, mushroom soup, both or neither. One of these people is selected at random. • \(T\) is the event that this person likes tomato soup. • \(M\) is the event that this person likes mushroom soup. You are given that \(\text{P}(T) = 0.55\), \(\text{P}(M) = 0.33\) and \(\text{P}(T|M) = 0.80\).
  1. Use this information to show that the events \(T\) and \(M\) are not independent. [1]
  2. Find \(\text{P}(T \cap M)\). [2]
  3. Draw a Venn diagram showing the events \(T\) and \(M\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
OCR MEI S1 Q4
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 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 \(\text{P}(W) = 0.14\), \(\text{P}(F) = 0.41\) and \(\text{P}(W \cap F) = 0.11\).
  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 \(\text{P}(W|F)\) and explain what this probability represents. [3]
OCR MEI S1 Q6
4 marks Moderate -0.8
The table shows all the possible products of the scores on two fair four-sided dice. \includegraphics{figure_6}
  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 Q7
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_7} 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]
AQA S2 2010 June Q1
9 marks Moderate -0.3
Judith, the village postmistress, believes that, since moving the post office counter into the local pharmacy, the mean daily number of customers that she serves has increased from \(79\). In order to investigate her belief, she counts the number of customers that she serves on \(12\) randomly selected days, with the following results. \(88 \quad 81 \quad 84 \quad 89 \quad 90 \quad 77 \quad 72 \quad 80 \quad 82 \quad 81 \quad 75 \quad 85\) Stating a necessary distributional assumption, test Judith's belief at the \(5\%\) level of significance. [9 marks]
AQA S2 2010 June Q2
8 marks Standard +0.3
It is claimed that a new drug is effective in the prevention of sickness in holiday-makers. A sample of \(100\) holiday-makers was surveyed, with the following results.
SicknessNo sicknessTotal
Drug taken245680
No drug taken11920
Total3565100
Assuming that the \(100\) holiday-makers are a random sample, use a \(\chi^2\) test, at the \(5\%\) level of significance, to investigate the claim. [8 marks]
AQA S2 2010 June Q3
10 marks Moderate -0.8
The continuous random variable \(X\) has a rectangular distribution defined by $$f(x) = \begin{cases} k & -3k \leqslant x \leqslant k \\ 0 & \text{otherwise} \end{cases}$$
    1. Sketch the graph of f. [2 marks]
    2. Hence show that \(k = \frac{1}{2}\). [2 marks]
  2. Find the exact numerical values for the mean and the standard deviation of \(X\). [3 marks]
    1. Find \(\mathrm{P}\left(X \geqslant -\frac{1}{4}\right)\). [2 marks]
    2. Write down the value of \(\mathrm{P}\left(X \neq -\frac{1}{4}\right)\). [1 mark]
AQA S2 2010 June Q4
5 marks Standard +0.3
The error, \(X\) °C, made in measuring a patient's temperature at a local doctors' surgery may be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). The errors, \(x\) °C, made in measuring the temperature of each of a random sample of \(10\) patients are summarised below. $$\sum x = 0.35 \quad \text{and} \quad \sum(x - \bar{x})^2 = 0.12705$$ Construct a \(99\%\) confidence interval for \(\mu\), giving the limits to three decimal places. [5 marks]
AQA S2 2010 June Q5
13 marks Standard +0.3
The number of telephone calls received, during an \(8\)-hour period, by an IT company that request an urgent visit by an engineer may be modelled by a Poisson distribution with a mean of \(7\).
  1. Determine the probability that, during a given \(8\)-hour period, the number of telephone calls received that request an urgent visit by an engineer is:
    1. at most \(5\); [1 mark]
    2. exactly \(7\); [2 marks]
    3. at least \(5\) but fewer than \(10\). [3 marks]
  2. Write down the distribution for the number of telephone calls received each hour that request an urgent visit by an engineer. [1 mark]
  3. The IT company has \(4\) engineers available for urgent visits and it may be assumed that each of these engineers takes exactly \(1\) hour for each such visit. At \(10\)am on a particular day, all \(4\) engineers are available for urgent visits.
    1. State the maximum possible number of telephone calls received between \(10\)am and \(11\)am that request an urgent visit and for which an engineer is immediately available. [1 mark]
    2. Calculate the probability that at \(11\)am an engineer will not be immediately available to make an urgent visit. [4 marks]
  4. Give a reason why a Poisson distribution may not be a suitable model for the number of telephone calls per hour received by the IT company that request an urgent visit by an engineer. [1 mark]
AQA S2 2010 June Q6
18 marks Standard +0.3
  1. The number of strokes, \(R\), taken by the members of Duffers Golf Club to complete the first hole may be modelled by the following discrete probability distribution.
    \(r\)\(\leqslant 2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(\geqslant 9\)
    \(\mathrm{P}(R = r)\)\(0\)\(0.1\)\(0.2\)\(0.3\)\(0.25\)\(0.1\)\(0.05\)\(0\)
    1. Determine the probability that a member, selected at random, takes at least \(5\) strokes to complete the first hole. [1 mark]
    2. Calculate \(\mathrm{E}(R)\). [2 marks]
    3. Show that \(\mathrm{Var}(R) = 1.66\). [4 marks]
  2. The number of strokes, \(S\), taken by the members of Duffers Golf Club to complete the second hole may be modelled by the following discrete probability distribution.
    \(s\)\(\leqslant 2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(\geqslant 9\)
    \(\mathrm{P}(S = s)\)\(0\)\(0.15\)\(0.4\)\(0.3\)\(0.1\)\(0.03\)\(0.02\)\(0\)
    Assuming that \(R\) and \(S\) are independent:
    1. show that \(\mathrm{P}(R + S \leqslant 8) = 0.24\); [5 marks]
    2. calculate the probability that, when \(5\) members are selected at random, at least \(4\) of them complete the first two holes in fewer than \(9\) strokes; [3 marks]
    3. calculate \(\mathrm{P}(R = 4 \mid R + S \leqslant 8)\). [3 marks]
AQA S2 2010 June Q7
12 marks Standard +0.3
The random variable \(X\) has probability density function defined by $$f(x) = \begin{cases} \frac{1}{2} & 0 \leqslant x \leqslant 1 \\ \frac{1}{18}(x - 4)^2 & 1 \leqslant x \leqslant 4 \\ 0 & \text{otherwise} \end{cases}$$
  1. State values for the median and the lower quartile of \(X\). [2 marks]
  2. Show that, for \(1 \leqslant x \leqslant 4\), the cumulative distribution function, \(\mathrm{F}(x)\), of \(X\) is given by $$\mathrm{F}(x) = 1 + \frac{1}{54}(x - 4)^3$$ (You may assume that \(\int (x - 4)^2 \, dx = \frac{1}{3}(x - 4)^3 + c\).) [4 marks]
  3. Determine \(\mathrm{P}(2 \leqslant X \leqslant 3)\). [2 marks]
    1. Show that \(q\), the upper quartile of \(X\), satisfies the equation \((q - 4)^3 = -13.5\). [3 marks]
    2. Hence evaluate \(q\) to three decimal places. [1 mark]
AQA S2 2016 June Q1
13 marks Standard +0.3
The water in a pond contains three different species of a spherical green algae: Volvox globator, at an average rate of 4.5 spheres per 1 cm³; Volvox aureus, at an average rate of 2.3 spheres per 1 cm³; Volvox tertius, at an average rate of 1.2 spheres per 1 cm³. Individual Volvox spheres may be considered to occur randomly and independently of all other Volvox spheres. Random samples of water are collected from this pond. Find the probability that:
  1. a 1 cm³ sample contains no more than 5 Volvox globator spheres; [1 mark]
  2. a 1 cm³ sample contains at least 2 Volvox aureus spheres; [3 marks]
  3. a 5 cm³ sample contains more than 8 but fewer than 12 Volvox tertius spheres; [3 marks]
  4. a 0.1 cm³ sample contains a total of exactly 2 Volvox spheres; [3 marks]
  5. a 1 cm³ sample contains at least 1 sphere of each of the three different species of algae. [3 marks]
AQA S2 2016 June Q2
4 marks Moderate -0.3
A normally distributed variable, \(X\), has unknown mean \(\mu\) and unknown standard deviation \(\sigma\). A sample of 10 values of \(X\) was taken. From these 10 values, a 95% confidence interval for \(\mu\) was calculated to be $$(30.47, 32.93)$$ Use this confidence interval to find unbiased estimates for \(\mu\) and \(\sigma^2\). [4 marks]
AQA S2 2016 June Q3
13 marks Moderate -0.8
Members of a library may borrow up to 6 books. Past experience has shown that the number of books borrowed, \(X\), follows the distribution shown in the table.
\(x\)0123456
P(X = x)00.190.260.200.130.070.15
  1. Find the probability that a member borrows more than 3 books. [1 mark]
  2. Assume that the numbers of books borrowed by two particular members are independent. Find the probability that one of these members borrows more than 3 books and the other borrows fewer than 3 books. [3 marks]
  3. Show that the mean of \(X\) is 3.08, and calculate the variance of \(X\). [4 marks]
  4. One of the library staff notices that the values of the mean and the variance of \(X\) are similar and suggests that a Poisson distribution could be used to model \(X\). Without further calculations, give two reasons why a Poisson distribution would not be suitable to model \(X\). [2 marks]
  5. The library introduces a fee of 10 pence for each book borrowed. Assuming that the probabilities do not change, calculate:
    1. the mean amount that will be paid by a member;
    2. the standard deviation of the amount that will be paid by a member.
    [3 marks]
AQA S2 2016 June Q4
7 marks Moderate -0.8
A digital thermometer measures temperatures in degrees Celsius. The thermometer rounds down the actual temperature to one decimal place, so that, for example, 36.23 and 36.28 are both shown as 36.2. The error, \(X\) °C, resulting from this rounding down can be modelled by a rectangular distribution with the following probability density function. $$f(x) = \begin{cases} k & 0 \leqslant x \leqslant 0.1 \\ 0 & \text{otherwise} \end{cases}$$
  1. State the value of \(k\). [1 mark]
  2. Find the probability that the error resulting from this rounding down is greater than 0.03 °C. [1 mark]
    1. State the value for E(\(X\)).
    2. Use integration to find the value for E(\(X^2\)).
    3. Hence find the value for the standard deviation of \(X\).
    [5 marks]
AQA S2 2016 June Q5
13 marks Standard +0.3
A car manufacturer keeps a record of how many of the new cars that it has sold experience mechanical problems during the first year. The manufacturer also records whether the cars have a petrol engine or a diesel engine. Data for a random sample of 250 cars are shown in the table.
Problems during first 3 monthsProblems during first year but after first 3 monthsNo problems during first yearTotal
Petrol engine1035170215
Diesel engine482335
Total1443193250
  1. Use a \(\chi^2\)-test to investigate, at the 10% significance level, whether there is an association between the mechanical problems experienced by a new car from this manufacturer and the type of engine. [11 marks]
  2. Arisa is planning to buy a new car from this manufacturer. She would prefer to buy a car with a diesel engine, but a friend has told her that cars with diesel engines experience more mechanical problems. Based on your answer to part (a), state, with a reason, the advice that you would give to Arisa. [2 marks]
AQA S2 2016 June Q6
16 marks Standard +0.3
Gerald is a scientist who studies sand lizards. He believes that sand lizards on islands are, on average, shorter than those on the mainland. The population of sand lizards on the mainland has a mean length of 18.2 cm and a standard deviation of 1.8 cm. Gerald visited three islands, A, B and C, and measured the length, \(X\) centimetres, of each of a sample of \(n\) sand lizards on each island. The samples may be regarded as random. The data are shown in the table.
Island\(\sum x\)\(n\)
A1384.578
B116.97
C394.620
  1. Carry out a hypothesis test to investigate whether the data from Island A provide support for Gerald's belief at the 2% significance level. Assume that the standard deviation of the lengths of sand lizards on Island A is 1.8 cm. [7 marks]
  2. For Island B, it is also given that $$\sum(x - \bar{x})^2 = 22.64$$
    1. Construct a 95% confidence interval for \(\mu_B\), where \(\mu_B\) centimetres is the mean length of sand lizards on Island B. Assume that the lengths of sand lizards on Island B are normally distributed with unknown standard deviation.
    2. Comment on whether your confidence interval provides support for Gerald's belief.
    [7 marks]
  3. Comment on whether the data from Island C provide support for Gerald's belief. [2 marks]
AQA S2 2016 June Q7
9 marks Standard +0.3
The continuous random variable \(X\) has a cumulative distribution function F(\(x\)), where $$\text{F}(x) = \begin{cases} 0 & x < 1 \\ \frac{1}{4}(x - 1) & 1 \leqslant x < 4 \\ \frac{1}{16}(12x - x^2 - 20) & 4 \leqslant x \leqslant 6 \\ 1 & x > 6 \end{cases}$$
  1. Sketch the probability density function, f(\(x\)), on the grid below. [5 marks]
  2. Find the mean value of \(X\). [4 marks]
Edexcel S2 Q1
4 marks Easy -2.0
  1. Explain why it is often useful to take samples as a means of obtaining information. [2 marks]
  2. Briefly define the term sampling frame. [1 mark]
  3. Suggest a suitable sampling frame for a sample survey on a proposal to install speed humps on a road. [1 mark]