Questions — CAIE (7646 questions)

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CAIE S1 2023 March Q7
12 marks Standard +0.3
  1. Find the number of different arrangements of the 9 letters in the word DELIVERED in which the three Es are together and the two Ds are not next to each other. [4]
  2. Find the probability that a randomly chosen arrangement of the 9 letters in the word DELIVERED has exactly 4 letters between the two Ds. [5]
Five letters are selected from the 9 letters in the word DELIVERED.
  1. Find the number of different selections if the 5 letters include at least one D and at least one E. [3]
CAIE S1 2002 June Q1
4 marks Easy -1.2
Events \(A\) and \(B\) are such that \(\text{P}(A) = 0.3\), \(\text{P}(B) = 0.8\) and \(\text{P}(A \text{ and } B) = 0.4\). State, giving a reason in each case, whether events \(A\) and \(B\) are
  1. independent, [2]
  2. mutually exclusive. [2]
CAIE S1 2002 June Q2
6 marks Easy -1.2
The manager of a company noted the times spent in 80 meetings. The results were as follows.
Time (\(t\) minutes)\(0 < t \leq 15\)\(15 < t \leq 30\)\(30 < t \leq 60\)\(60 < t \leq 90\)\(90 < t \leq 120\)
Number of meetings4724387
Draw a cumulative frequency graph and use this to estimate the median time and the interquartile range. [6]
CAIE S1 2002 June Q3
7 marks Moderate -0.8
A fair cubical die with faces numbered 1, 1, 1, 2, 3, 4 is thrown and the score noted. The area \(A\) of a square of side equal to the score is calculated, so, for example, when the score on the die is 3, the value of \(A\) is 9.
  1. Draw up a table to show the probability distribution of \(A\). [3]
  2. Find \(\text{E}(A)\) and \(\text{Var}(A)\). [4]
CAIE S1 2002 June Q4
7 marks Moderate -0.8
  1. In a spot check of the speeds \(x \text{ km h}^{-1}\) of 30 cars on a motorway, the data were summarised by \(\Sigma(x - 110) = -47.2\) and \(\Sigma(x - 110)^2 = 5460\). Calculate the mean and standard deviation of these speeds. [4]
  2. On another day the mean speed of cars on the motorway was found to be \(107.6 \text{ km h}^{-1}\) and the standard deviation was \(13.8 \text{ km h}^{-1}\). Assuming these speeds follow a normal distribution and that the speed limit is \(110 \text{ km h}^{-1}\), find what proportion of cars exceed the speed limit. [3]
CAIE S1 2002 June Q5
8 marks Moderate -0.3
The digits of the number 1223678 can be rearranged to give many different 7-digit numbers. Find how many different 7-digit numbers can be made if
  1. there are no restrictions on the order of the digits, [2]
  2. the digits 1, 3, 7 (in any order) are next to each other, [3]
  3. these 7-digit numbers are even. [3]
CAIE S1 2002 June Q6
8 marks Standard +0.3
  1. In a normal distribution with mean \(\mu\) and standard deviation \(\sigma\), \(\text{P}(X > 3.6) = 0.5\) and \(\text{P}(X > 2.8) = 0.6554\). Write down the value of \(\mu\), and calculate the value of \(\sigma\). [4]
  2. If four observations are taken at random from this distribution, find the probability that at least two observations are greater than 2.8. [4]
CAIE S1 2002 June Q7
10 marks Moderate -0.3
  1. A garden shop sells polyanthus plants in boxes, each box containing the same number of plants. The number of plants per box which produce yellow flowers has a binomial distribution with mean 11 and variance 4.95.
    1. Find the number of plants per box. [4]
    2. Find the probability that a box contains exactly 12 plants which produce yellow flowers. [2]
  2. Another garden shop sells polyanthus plants in boxes of 100. The shop's advertisement states that the probability of any polyanthus plant producing a pink flower is 0.3. Use a suitable approximation to find the probability that a box contains fewer than 35 plants which produce pink flowers. [4]
CAIE S1 2010 June Q1
5 marks Moderate -0.8
The times in minutes for seven students to become proficient at a new computer game were measured. The results are shown below. $$15 \quad 10 \quad 48 \quad 10 \quad 19 \quad 14 \quad 16$$
  1. Find the mean and standard deviation of these times. [2]
  2. State which of the mean, median or mode you consider would be most appropriate to use as a measure of central tendency to represent the data in this case. [1]
  3. For each of the two measures of average you did not choose in part (ii), give a reason why you consider it inappropriate. [2]
CAIE S1 2010 June Q2
5 marks Moderate -0.8
The lengths of new pencils are normally distributed with mean 11 cm and standard deviation 0.095 cm.
  1. Find the probability that a pencil chosen at random has a length greater than 10.9 cm. [2]
  2. Find the probability that, in a random sample of 6 pencils, at least two have lengths less than 10.9 cm. [3]
CAIE S1 2010 June Q3
6 marks Moderate -0.8
\includegraphics{figure_3} The birth weights of random samples of 900 babies born in country \(A\) and 900 babies born in country \(B\) are illustrated in the cumulative frequency graphs. Use suitable data from these graphs to compare the central tendency and spread of the birth weights of the two sets of babies. [6]
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]