Questions — CAIE S1 (789 questions)

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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]