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CAIE S1 2010 June Q4
6 marks Standard +0.3
The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\).
  1. Given that \(5\sigma = 3\mu\), find \(\mathrm{P}(X < 2\mu)\). [3]
  2. With a different relationship between \(\mu\) and \(\sigma\), it is given that \(\mathrm{P}(X < \frac{4\mu}{3}) = 0.8524\). Express \(\mu\) in terms of \(\sigma\). [3]
CAIE S1 2010 June Q5
8 marks Moderate -0.8
Two fair twelve-sided dice with sides marked 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 are thrown, and the numbers on the sides which land face down are noted. Events \(Q\) and \(R\) are defined as follows. \(Q\): the product of the two numbers is 24. \(R\): both of the numbers are greater than 8.
  1. Find \(\mathrm{P}(Q)\). [2]
  2. Find \(\mathrm{P}(R)\). [2]
  3. Are events \(Q\) and \(R\) exclusive? Justify your answer. [2]
  4. Are events \(Q\) and \(R\) independent? Justify your answer. [2]
CAIE S1 2010 June Q6
10 marks Moderate -0.3
A small farm has 5 ducks and 2 geese. Four of these birds are to be chosen at random. The random variable \(X\) represents the number of geese chosen.
  1. Draw up the probability distribution of \(X\). [3]
  2. Show that \(\mathrm{E}(X) = \frac{8}{7}\) and calculate \(\mathrm{Var}(X)\). [3]
  3. When the farmer's dog is let loose, it chases either the ducks with probability \(\frac{3}{5}\) or the geese with probability \(\frac{2}{5}\). If the dog chases the ducks there is a probability of \(\frac{1}{10}\) that they will attack the dog. If the dog chases the geese there is a probability of \(\frac{1}{4}\) that they will attack the dog. Given that the dog is not attacked, find the probability that it was chasing the geese. [4]
CAIE S1 2010 June Q7
10 marks Moderate -0.8
Nine cards, each of a different colour, are to be arranged in a line.
  1. How many different arrangements of the 9 cards are possible? [1]
The 9 cards include a pink card and a green card.
  1. How many different arrangements do not have the pink card next to the green card? [3]
Consider all possible choices of 3 cards from the 9 cards with the 3 cards being arranged in a line.
  1. How many different arrangements in total of 3 cards are possible? [2]
  2. How many of the arrangements of 3 cards in part (iii) contain the pink card? [2]
  3. How many of the arrangements of 3 cards in part (iii) do not have the pink card next to the green card? [2]
CAIE S1 2015 June Q1
3 marks Moderate -0.5
A fair die is thrown 10 times. Find the probability that the number of sixes obtained is between 3 and 5 inclusive. [3]
CAIE S1 2015 June Q2
5 marks Moderate -0.8
120 people were asked to read an article in a newspaper. The times taken, to the nearest second, by the people to read the article are summarised in the following table.
Time (seconds)1 -- 2526 -- 3536 -- 4546 -- 5556 -- 90
Number of people424383420
Calculate estimates of the mean and standard deviation of the reading times. [5]
CAIE S1 2015 June Q3
6 marks Easy -1.2
\includegraphics{figure_3} In an open-plan office there are 88 computers. The times taken by these 88 computers to access a particular web page are represented in the cumulative frequency diagram.
  1. On graph paper draw a box-and-whisker plot to summarise this information. [4]
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. Show that there are no outliers. [2]
CAIE S1 2015 June Q4
7 marks Moderate -0.3
[diagram]
Nikita goes shopping to buy a birthday present for her mother. She buys either a scarf, with probability 0.3, or a handbag. The probability that her mother will like the choice of scarf is 0.72. The probability that her mother will like the choice of handbag is \(x\). This information is shown on the tree diagram. The probability that Nikita's mother likes the present that Nikita buys is 0.783.
  1. Find \(x\). [3]
  2. Given that Nikita's mother does not like her present, find the probability that the present is a scarf. [4]
CAIE S1 2015 June Q5
8 marks Moderate -0.8
A box contains 5 discs, numbered 1, 2, 4, 6, 7. William takes 3 discs at random, without replacement, and notes the numbers on the discs.
  1. Find the probability that the numbers on the 3 discs are two even numbers and one odd number. [3]
The smallest of the numbers on the 3 discs taken is denoted by the random variable \(S\).
  1. By listing all possible selections (126, 246 and so on) draw up the probability distribution table for \(S\). [5]
CAIE S1 2015 June Q6
9 marks Moderate -0.8
  1. Find the number of different ways the 7 letters of the word BANANAS can be arranged
    1. if the first letter is N and the last letter is B, [3]
    2. if all the letters A are next to each other. [3]
  2. Find the number of ways of selecting a group of 9 people from 14 if two particular people cannot both be in the group together. [3]
CAIE S1 2015 June Q7
12 marks Moderate -0.3
  1. Once a week Zak goes for a run. The time he takes, in minutes, has a normal distribution with mean 35.2 and standard deviation 4.7.
    1. Find the expected number of days during a year (52 weeks) for which Zak takes less than 30 minutes for his run. [4]
    2. The probability that Zak's time is between 35.2 minutes and \(t\) minutes, where \(t > 35.2\), is 0.148. Find the value of \(t\). [3]
  2. The random variable \(X\) has the distribution \(\text{N}(\mu, \sigma^2)\). It is given that \(\text{P}(X < 7) = 0.2119\) and \(\text{P}(X < 10) = 0.6700\). Find the values of \(\mu\) and \(\sigma\). [5]
CAIE S1 2014 November Q1
3 marks Easy -1.2
The 50 members of a club include both the club president and the club treasurer. All 50 members want to go on a coach tour, but the coach only has room for 45 people. In how many ways can 45 members be chosen if both the club president and the club treasurer must be included? [3]
CAIE S1 2014 November Q2
6 marks Moderate -0.3
Find the number of different ways that 6 boys and 4 girls can stand in a line if
  1. all 6 boys stand next to each other, [3]
  2. no girl stands next to another girl. [3]
CAIE S1 2014 November Q3
7 marks Standard +0.3
  1. Four fair six-sided dice, each with faces marked 1, 2, 3, 4, 5, 6, are thrown. Find the probability that the numbers shown on the four dice add up to 5. [3]
  2. Four fair six-sided dice, each with faces marked 1, 2, 3, 4, 5, 6, are thrown on 7 occasions. Find the probability that the numbers shown on the four dice add up to 5 on exactly 1 or 2 of the 7 occasions. [4]
CAIE S1 2014 November Q4
8 marks Moderate -0.8
Sharik attempts a multiple choice revision question on-line. There are 3 suggested answers, one of which is correct. When Sharik chooses an answer the computer indicates whether the answer is right or wrong. Sharik first chooses one of the three suggested answers at random. If this answer is wrong he has a second try, choosing an answer at random from the remaining 2. If this answer is also wrong Sharik then chooses the remaining answer, which must be correct.
  1. Draw a fully labelled tree diagram to illustrate the various choices that Sharik can make until the computer indicates that he has answered the question correctly. [4]
  2. The random variable \(X\) is the number of attempts that Sharik makes up to and including the one that the computer indicates is correct. Draw up the probability distribution table for \(X\) and find E\((X)\). [4]
CAIE S1 2014 November Q5
8 marks Moderate -0.8
  1. The time, \(X\) hours, for which people sleep in one night has a normal distribution with mean 7.15 hours and standard deviation 0.88 hours.
    1. Find the probability that a randomly chosen person sleeps for less than 8 hours in a night. [2]
    2. Find the value of \(q\) such that P\((X < q) = 0.75\). [3]
  2. The random variable \(Y\) has the distribution N\((\mu, \sigma^2)\), where \(2\sigma = 3\mu\) and \(\mu \neq 0\). Find P\((Y > 4\mu)\). [3]
CAIE S1 2014 November Q6
9 marks Easy -1.2
On a certain day in spring, the heights of 200 daffodils are measured, correct to the nearest centimetre. The frequency distribution is given below.
Height (cm)\(4 - 10\)\(11 - 15\)\(16 - 20\)\(21 - 25\)\(26 - 30\)
Frequency2232784028
  1. Draw a cumulative frequency graph to illustrate the data. [4]
  2. 28\% of these daffodils are of height \(h\) cm or more. Estimate \(h\). [2]
  3. You are given that the estimate of the mean height of these daffodils, calculated from the table, is 18.39 cm. Calculate an estimate of the standard deviation of the heights of these daffodils. [3]
CAIE S1 2014 November Q7
9 marks Standard +0.3
In Marumbo, three quarters of the adults own a cell phone.
  1. A random sample of 8 adults from Marumbo is taken. Find the probability that the number of adults who own a cell phone is between 4 and 6 inclusive. [3]
  2. A random sample of 160 adults from Marumbo is taken. Use an approximation to find the probability that more than 114 of them own a cell phone. [5]
  3. Justify the use of your approximation in part (ii). [1]
Edexcel S1 2023 June Q1
8 marks Moderate -0.8
The histogram shows the distances, in km, that 274 people travel to work. \includegraphics{figure_1} Given that 60 of these people travel between 10km and 20km to work, estimate
  1. the number of people who travel between 22km and 45km to work, [3]
  2. the median distance travelled to work by these 274 people, [2]
  3. the mean distance travelled to work by these 274 people. [3]
Edexcel S1 2023 June Q2
13 marks Moderate -0.3
Two students, Olive and Shan, collect data on the weight, \(w\) grams, and the tail length, \(t\) cm, of 15 mice. Olive summarised the data as follows \(S_tt = 5.3173\) \quad \(\sum w^2 = 6089.12\) \quad \(\sum tw = 2304.53\) \quad \(\sum w = 297.8\) \quad \(\sum t = 114.8\)
  1. Calculate the value of \(S_{ww}\) and the value of \(S_{tw}\) [3]
  2. Calculate the value of the product moment correlation coefficient between \(w\) and \(t\) [2]
  3. Show that the equation of the regression line of \(w\) on \(t\) can be written as $$w = -16.7 + 4.77t$$ [3]
  4. Give an interpretation of the gradient of the regression line. [1]
  5. Explain why it would not be appropriate to use the regression line in part (c) to estimate the weight of a mouse with a tail length of 2cm. [2]
Shan decided to code the data using \(x = t - 6\) and \(y = \frac{w}{2} - 5\)
  1. Write down the value of the product moment correlation coefficient between \(x\) and \(y\) [1]
  2. Write down an equation of the regression line of \(y\) on \(x\) You do not need to simplify your equation. [1]
Edexcel S1 2023 June Q3
9 marks Moderate -0.8
Jim records the length, \(l\) mm, of 81 salmon. The data are coded using \(x = l - 600\) and the following summary statistics are obtained. $$n = 81 \quad \sum x = 3711 \quad \sum x^2 = 475181$$
  1. Find the mean length of these salmon. [3]
  2. Find the variance of the lengths of these salmon. [2]
The weight, \(w\) grams, of each of the 81 salmon is recorded to the nearest gram. The recorded results for the 81 salmon are summarised in the box plot below. \includegraphics{figure_2}
  1. Find the maximum number of salmon that have weights in the interval $$4600 < w \leqslant 7700$$ [1]
Raj says that the box plot is incorrect as Jim has not included outliers. For these data an outlier is defined as a value that is more than \(1.5 \times\) IQR above the upper quartile \quad or \quad \(1.5 \times\) IQR below the lower quartile
  1. Show that there are no outliers. [3]
Edexcel S1 2023 June Q4
9 marks Moderate -0.8
A bag contains a large number of coloured counters. Each counter is labelled A, B or C 30% of the counters are labelled A 45% of the counters are labelled B The rest of the counters are labelled C It is known that 2% of the counters labelled A are red 4% of the counters labelled B are red 6% of the counters labelled C are red One counter is selected at random from the bag.
  1. Complete the tree diagram on the opposite page to illustrate this information. [2]
  2. Calculate the probability that the counter is labelled A and is not red. [2]
  3. Calculate the probability that the counter is red. [2]
  4. Given that the counter is red, find the probability that it is labelled C [3]
\includegraphics{figure_3}
Edexcel S1 2023 June Q5
13 marks Standard +0.3
A discrete random variable \(Y\) has probability function $$\mathrm{P}(Y = y) = \begin{cases} k(3 - y) & y = 1, 2 \\ k(y^2 - 8) & y = 3, 4, 5 \\ k & y = 6 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac{1}{30}\) [2]
Find the exact value of
  1. P\((1 < Y \leqslant 4)\) [2]
  2. E\((Y)\) [2]
The random variable \(X = 15 - 2Y\)
  1. Calculate P\((Y \geqslant X)\) [3]
  2. Calculate Var\((X)\) [4]
Edexcel S1 2023 June Q6
9 marks Moderate -0.3
Three events \(A\), \(B\) and \(C\) are such that $$\mathrm{P}(A) = 0.1 \quad \mathrm{P}(B|A) = 0.3 \quad \mathrm{P}(A \cup B) = 0.25 \quad \mathrm{P}(C) = 0.5$$ Given that \(A\) and \(C\) are mutually exclusive
  1. find P\((A \cup C)\) [1]
  2. Show that P\((B) = 0.18\) [3]
Given also that \(B\) and \(C\) are independent,
  1. draw a Venn diagram to represent the events \(A\), \(B\) and \(C\) and the probabilities associated with each region. [5]
Edexcel S1 2023 June Q7
14 marks Standard +0.3
A machine squeezes apples to extract their juice. The volume of juice, \(J\) ml, extracted from 1 kg of apples is modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\) Given that \(\mu = 500\) and \(\sigma = 25\) use standardisation to
    1. show that P\((J > 510) = 0.3446\) [2]
    2. calculate the value of \(d\) such that P\((J > d) = 0.9192\) [3]
Zen randomly selects 5 bags each containing 1 kg of apples and records the volume of juice extracted from each bag of apples.
  1. Calculate the probability that each of the 5 bags of apples produce less than 510ml of juice. [2]
Following adjustments to the machine, the volume of juice, \(R\) ml, extracted from 1 kg of apples is such that \(\mu = 520\) and \(\sigma = k\) Given that P\((R < r) = 0.15\) and P\((R > 3r - 800) = 0.005\)
  1. find the value of \(r\) and the value of \(k\) [7]