Questions — CAIE S1 (785 questions)

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CAIE S1 2012 November Q3
3 The table summarises the times that 112 people took to travel to work on a particular day.
Time to travel to
work \(( t\) minutes \()\)
\(0 < t \leqslant 10\)\(10 < t \leqslant 15\)\(15 < t \leqslant 20\)\(20 < t \leqslant 25\)\(25 < t \leqslant 40\)\(40 < t \leqslant 60\)
Frequency191228221813
  1. State which time interval in the table contains the median and which time interval contains the upper quartile.
  2. On graph paper, draw a histogram to represent the data.
  3. Calculate an estimate of the mean time to travel to work.
CAIE S1 2012 November Q4
4 The mean of a certain normally distributed variable is four times the standard deviation. The probability that a randomly chosen value is greater than 5 is 0.15 .
  1. Find the mean and standard deviation.
  2. 200 values of the variable are chosen at random. Find the probability that at least 160 of these values are less than 5 .
CAIE S1 2012 November Q5
5
  1. A team of 3 boys and 3 girls is to be chosen from a group of 12 boys and 9 girls to enter a competition. Tom and Henry are two of the boys in the group. Find the number of ways in which the team can be chosen if Tom and Henry are either both in the team or both not in the team.
  2. The back row of a cinema has 12 seats, all of which are empty. A group of 8 people, including Mary and Frances, sit in this row. Find the number of different ways they can sit in these 12 seats if
    1. there are no restrictions,
    2. Mary and Frances do not sit in seats which are next to each other,
    3. all 8 people sit together with no empty seats between them.
CAIE S1 2012 November Q6
6 A fair tetrahedral die has four triangular faces, numbered \(1,2,3\) and 4 . The score when this die is thrown is the number on the face that the die lands on. This die is thrown three times. The random variable \(X\) is the sum of the three scores.
  1. Show that \(\mathrm { P } ( X = 9 ) = \frac { 10 } { 64 }\).
  2. Copy and complete the probability distribution table for \(X\).
    \(x\)3456789101112
    \(\mathrm { P } ( X = x )\)\(\frac { 1 } { 64 }\)\(\frac { 3 } { 64 }\)\(\frac { 12 } { 64 }\)
  3. Event \(R\) is 'the sum of the three scores is 9 '. Event \(S\) is 'the product of the three scores is 16 '. Determine whether events \(R\) and \(S\) are independent, showing your working.
CAIE S1 2012 November Q1
1 In a normal distribution with mean 9.3, the probability of a randomly chosen value being greater than 5.6 is 0.85 . Find the standard deviation.
CAIE S1 2012 November Q2
2 The discrete random variable \(X\) has the following probability distribution.
\(x\)- 3024
\(\mathrm { P } ( X = x )\)\(p\)\(q\)\(r\)0.4
Given that \(\mathrm { E } ( X ) = 2.3\) and \(\operatorname { Var } ( X ) = 3.01\), find the values of \(p , q\) and \(r\).
CAIE S1 2012 November Q3
3 Ronnie obtained data about the gross domestic product (GDP) and the birth rate for 170 countries. He classified each GDP and each birth rate as either 'low', 'medium' or 'high'. The table shows the number of countries in each category.
Birth rate
\cline { 3 - 5 } \multicolumn{2}{|c|}{}LowMediumHigh
\multirow{3}{*}{GDP}Low3545
\cline { 2 - 5 }Medium204212
\cline { 2 - 5 }High3580
One of these countries is chosen at random.
  1. Find the probability that the country chosen has a medium GDP.
  2. Find the probability that the country chosen has a low birth rate, given that it does not have a medium GDP.
  3. State with a reason whether or not the events 'the country chosen has a high GDP' and 'the country chosen has a high birth rate' are exclusive. One country is chosen at random from those countries which have a medium GDP and then a different country is chosen at random from those which have a medium birth rate.
  4. Find the probability that both countries chosen have a medium GDP and a medium birth rate.
CAIE S1 2012 November Q4
4 In a survey, the percentage of meat in a certain type of take-away meal was found. The results, to the nearest integer, for 193 take-away meals are summarised in the table.
Percentage of meat\(1 - 5\)\(6 - 10\)\(11 - 20\)\(21 - 30\)\(31 - 50\)
Frequency5967381811
  1. Calculate estimates of the mean and standard deviation of the percentage of meat in these take-away meals.
  2. Draw, on graph paper, a histogram to illustrate the information in the table.
CAIE S1 2012 November Q5
5 The random variable \(X\) is such that \(X \sim \mathrm {~N} ( 82,126 )\).
  1. A value of \(X\) is chosen at random and rounded to the nearest whole number. Find the probability that this whole number is 84 .
  2. Five independent observations of \(X\) are taken. Find the probability that at most one of them is greater than 87.
  3. Find the value of \(k\) such that \(\mathrm { P } ( 87 < X < k ) = 0.3\).
CAIE S1 2012 November Q6
6
  1. A chess team of 2 girls and 2 boys is to be chosen from the 7 girls and 6 boys in the chess club. Find the number of ways this can be done if 2 of the girls are twins and are either both in the team or both not in the team.
    1. The digits of the number 1244687 can be rearranged to give many different 7-digit numbers. How many of these 7 -digit numbers are even?
    2. How many different numbers between 20000 and 30000 can be formed using 5 different digits from the digits \(1,2,4,6,7,8\) ?
  3. Helen has some black tiles, some white tiles and some grey tiles. She places a single row of 8 tiles above her washbasin. Each tile she places is equally likely to be black, white or grey. Find the probability that there are no tiles of the same colour next to each other.
CAIE S1 2013 November Q1
1 It is given that \(X \sim \mathrm {~N} ( 30,49 ) , Y \sim \mathrm {~N} ( 30,16 )\) and \(Z \sim \mathrm {~N} ( 50,16 )\). On a single diagram, with the horizontal axis going from 0 to 70 , sketch three curves to represent the distributions of \(X , Y\) and \(Z\).
CAIE S1 2013 November Q2
5 marks
2 The people living in two towns, Mumbok and Bagville, are classified by age. The numbers in thousands living in each town are shown in the table below.
MumbokBagville
Under 18 years1535
18 to 60 years5595
Over 60 years2030
One of the towns is chosen. The probability of choosing Mumbok is 0.6 and the probability of choosing Bagville is 0.4 . Then a person is chosen at random from that town. Given that the person chosen is between 18 and 60 years old, find the probability that the town chosen was Mumbok. [5]
CAIE S1 2013 November Q3
3 Swati measured the lengths, \(x \mathrm {~cm}\), of 18 stick insects and found that \(\Sigma x ^ { 2 } = 967\). Given that the mean length is \(\frac { 58 } { 9 } \mathrm {~cm}\), find the values of \(\Sigma ( x - 5 )\) and \(\Sigma ( x - 5 ) ^ { 2 }\).
CAIE S1 2013 November Q4
4 The following are the house prices in thousands of dollars, arranged in ascending order, for 51 houses from a certain area.
253270310354386428433468472477485520520524526531535
536538541543546548549551554572583590605614638649652
666670682684690710725726731734745760800854863957986
  1. Draw a box-and-whisker plot to represent the data. An expensive house is defined as a house which has a price that is more than 1.5 times the interquartile range above the upper quartile.
  2. For the above data, give the prices of the expensive houses.
  3. Give one disadvantage of using a box-and-whisker plot rather than a stem-and-leaf diagram to represent this set of data.
CAIE S1 2013 November Q5
5 Lengths of a certain type of carrot have a normal distribution with mean 14.2 cm and standard deviation 3.6 cm .
  1. \(8 \%\) of carrots are shorter than \(c \mathrm {~cm}\). Find the value of \(c\).
  2. Rebekah picks 7 carrots at random. Find the probability that at least 2 of them have lengths between 15 and 16 cm .
CAIE S1 2013 November Q6
6 A shop has 7 different mountain bicycles, 5 different racing bicycles and 8 different ordinary bicycles on display. A cycling club selects 6 of these 20 bicycles to buy.
  1. How many different selections can be made if there must be no more than 3 mountain bicycles and no more than 2 of each of the other types of bicycle? The cycling club buys 3 mountain bicycles, 1 racing bicycle and 2 ordinary bicycles and parks them in a cycle rack, which has a row of 10 empty spaces.
  2. How many different arrangements are there in the cycle rack if the mountain bicycles are all together with no spaces between them, the ordinary bicycles are both together with no spaces between them and the spaces are all together?
  3. How many different arrangements are there in the cycle rack if the ordinary bicycles are at each end of the bicycles and there are no spaces between any of the bicycles?
CAIE S1 2013 November Q7
7 James has a fair coin and a fair tetrahedral die with four faces numbered 1, 2, 3, 4. He tosses the coin once and the die twice. The random variable \(X\) is defined as follows.
  • If the coin shows a head then \(X\) is the sum of the scores on the two throws of the die.
  • If the coin shows a tail then \(X\) is the score on the first throw of the die only.
    1. Explain why \(X = 1\) can only be obtained by throwing a tail, and show that \(\mathrm { P } ( X = 1 ) = \frac { 1 } { 8 }\).
    2. Show that \(\mathrm { P } ( X = 3 ) = \frac { 3 } { 16 }\).
    3. Copy and complete the probability distribution table for \(X\).
\(x\)12345678
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 8 }\)\(\frac { 3 } { 16 }\)\(\frac { 1 } { 8 }\)\(\frac { 1 } { 16 }\)\(\frac { 1 } { 32 }\)
Event \(Q\) is 'James throws a tail'. Event \(R\) is 'the value of \(X\) is 7'.
  • Determine whether events \(Q\) and \(R\) are exclusive. Justify your answer.
    •
    CAIE S1 2013 November Q1
    1 It is given that \(X \sim \mathrm {~N} \left( 1.5,3.2 ^ { 2 } \right)\). Find the probability that a randomly chosen value of \(X\) is less than - 2.4 .
    CAIE S1 2013 November Q2
    2 On Saturday afternoons Mohit goes shopping with probability 0.25, or goes to the cinema with probability 0.35 or stays at home. If he goes shopping the probability that he spends more than \(
    ) 50\( is 0.7 . If he goes to the cinema the probability that he spends more than \)\\( 50\) is 0.8 . If he stays at home he spends \(
    ) 10$ on a pizza.
    1. Find the probability that Mohit will go to the cinema and spend less than \(
      ) 50\(.
    2. Given that he spends less than \)\\( 50\), find the probability that he went to the cinema.
    CAIE S1 2013 November Q3
    3 The amount of fibre in a packet of a certain brand of cereal is normally distributed with mean 160 grams. 19\% of packets of cereal contain more than 190 grams of fibre.
    1. Find the standard deviation of the amount of fibre in a packet.
    2. Kate buys 12 packets of cereal. Find the probability that at least 1 of the packets contains more than 190 grams of fibre.
    CAIE S1 2013 November Q4
    4 The following histogram summarises the times, in minutes, taken by 190 people to complete a race.
    \includegraphics[max width=\textwidth, alt={}, center]{df246a50-157b-49f7-bba0-f9b86960b8b9-2_1210_1125_1251_513}
    1. Show that 75 people took between 200 and 250 minutes to complete the race.
    2. Calculate estimates of the mean and standard deviation of the times of the 190 people.
    3. Explain why your answers to part (ii) are estimates.
    CAIE S1 2013 November Q5
    5 On trains in the morning rush hour, each person is either a student with probability 0.36 , or an office worker with probability 0.22 , or a shop assistant with probability 0.29 or none of these.
    1. 8 people on a morning rush hour train are chosen at random. Find the probability that between 4 and 6 inclusive are office workers.
    2. 300 people on a morning rush hour train are chosen at random. Find the probability that between 31 and 49 inclusive are neither students nor office workers nor shop assistants.
    CAIE S1 2013 November Q6
    6 The 11 letters of the word REMEMBRANCE are arranged in a line.
    1. Find the number of different arrangements if there are no restrictions.
    2. Find the number of different arrangements which start and finish with the letter M .
    3. Find the number of different arrangements which do not have all 4 vowels ( \(\mathrm { E } , \mathrm { E } , \mathrm { A } , \mathrm { E }\) ) next to each other. 4 letters from the letters of the word REMEMBRANCE are chosen.
    4. Find the number of different selections which contain no Ms and no Rs and at least 2 Es.
    CAIE S1 2013 November Q7
    7 Rory has 10 cards. Four of the cards have a 3 printed on them and six of the cards have a 4 printed on them. He takes three cards at random, without replacement, and adds up the numbers on the cards.
    1. Show that P (the sum of the numbers on the three cards is \(11 ) = \frac { 1 } { 2 }\).
    2. Draw up a probability distribution table for the sum of the numbers on the three cards. Event \(R\) is 'the sum of the numbers on the three cards is 11 '. Event \(S\) is 'the number on the first card taken is a \(3 ^ { \prime }\).
    3. Determine whether events \(R\) and \(S\) are independent. Justify your answer.
    4. Determine whether events \(R\) and \(S\) are exclusive. Justify your answer.
    CAIE S1 2013 November Q1
    1 The distance of a student's home from college, correct to the nearest kilometre, was recorded for each of 55 students. The distances are summarised in the following table.
    Distance from college \(( \mathrm { km } )\)\(1 - 3\)\(4 - 5\)\(6 - 8\)\(9 - 11\)\(12 - 16\)
    Number of students18138124
    Dominic is asked to draw a histogram to illustrate the data. Dominic's diagram is shown below.
    \includegraphics[max width=\textwidth, alt={}, center]{d6836b62-75e7-410e-ab1e-83c391b85948-2_1225_1303_628_422} Give two reasons why this is not a correct histogram.