Questions S1 (1967 questions)

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CAIE S1 2014 June Q1
1 The petrol consumption of a certain type of car has a normal distribution with mean 24 kilometres per litre and standard deviation 4.7 kilometres per litre. Find the probability that the petrol consumption of a randomly chosen car of this type is between 21.6 kilometres per litre and 28.7 kilometres per litre.
CAIE S1 2014 June Q2
2 Lengths of a certain type of white radish are normally distributed with mean \(\mu \mathrm { cm }\) and standard deviation \(\sigma \mathrm { cm } .4 \%\) of these radishes are longer than 12 cm and \(32 \%\) are longer than 9 cm . Find \(\mu\) and \(\sigma\).
CAIE S1 2014 June Q3
3
  1. State three conditions which must be satisfied for a situation to be modelled by a binomial distribution. George wants to invest some of his monthly salary. He invests a certain amount of this every month for 18 months. For each month there is a probability of 0.25 that he will buy shares in a large company, there is a probability of 0.15 that he will buy shares in a small company and there is a probability of 0.6 that he will invest in a savings account.
  2. Find the probability that George will buy shares in a small company in at least 3 of these 18 months.
CAIE S1 2014 June Q4
4 A book club sends 6 paperback and 2 hardback books to Mrs Hunt. She chooses 4 of these books at random to take with her on holiday. The random variable \(X\) represents the number of paperback books she chooses.
  1. Show that the probability that she chooses exactly 2 paperback books is \(\frac { 3 } { 14 }\).
  2. Draw up the probability distribution table for \(X\).
  3. You are given that \(\mathrm { E } ( X ) = 3\). Find \(\operatorname { Var } ( X )\).
CAIE S1 2014 June Q5
5 Playground equipment consists of swings ( \(S\) ), roundabouts ( \(R\) ), climbing frames ( \(C\) ) and play-houses \(( P )\). The numbers of pieces of equipment in each of 3 playgrounds are as follows.
Playground \(X\)Playground \(Y\)Playground \(Z\)
\(3 S , 2 R , 4 P\)\(6 S , 3 R , 1 C , 2 P\)\(8 S , 3 R , 4 C , 1 P\)
Each day Nur takes her child to one of the playgrounds. The probability that she chooses playground \(X\) is \(\frac { 1 } { 4 }\). The probability that she chooses playground \(Y\) is \(\frac { 1 } { 4 }\). The probability that she chooses playground \(Z\) is \(\frac { 1 } { 2 }\). When she arrives at the playground, she chooses one piece of equipment at random.
  1. Find the probability that Nur chooses a play-house.
  2. Given that Nur chooses a climbing frame, find the probability that she chose playground \(Y\).
CAIE S1 2014 June Q6
6 Find the number of different ways in which all 8 letters of the word TANZANIA can be arranged so that
  1. all the letters A are together,
  2. the first letter is a consonant ( \(\mathrm { T } , \mathrm { N } , \mathrm { Z }\) ), the second letter is a vowel ( \(\mathrm { A } , \mathrm { I }\) ), the third letter is a consonant, the fourth letter is a vowel, and so on alternately. 4 of the 8 letters of the word TANZANIA are selected. How many possible selections contain
  3. exactly 1 N and 1 A ,
  4. exactly 1 N ?
CAIE S1 2014 June Q7
7 A typing test is taken by 111 people. The numbers of typing errors they make in the test are summarised in the table below.
Number of typing errors\(1 - 5\)\(6 - 20\)\(21 - 35\)\(36 - 60\)\(61 - 80\)
Frequency249211542
  1. Draw a histogram on graph paper to represent this information.
  2. Calculate an estimate of the mean number of typing errors for these 111 people.
  3. State which class contains the lower quartile and which class contains the upper quartile. Hence find the least possible value of the interquartile range.
CAIE S1 2014 June Q1
1 In a certain country \(12 \%\) of houses have solar heating. 19 houses are chosen at random. Find the probability that fewer than 4 houses have solar heating.
CAIE S1 2014 June Q2
2 A school club has members from 3 different year-groups: Year 1, Year 2 and Year 3. There are 7 members from Year 1, 2 members from Year 2 and 2 members from Year 3. Five members of the club are selected. Find the number of possible selections that include at least one member from each year-group.
CAIE S1 2014 June Q3
3 Roger and Andy play a tennis match in which the first person to win two sets wins the match. The probability that Roger wins the first set is 0.6 . For sets after the first, the probability that Roger wins the set is 0.7 if he won the previous set, and is 0.25 if he lost the previous set. No set is drawn.
  1. Find the probability that there is a winner of the match after exactly two sets.
  2. Find the probability that Andy wins the match given that there is a winner of the match after exactly two sets.
CAIE S1 2014 June Q4
4 Coin \(A\) is weighted so that the probability of throwing a head is \(\frac { 2 } { 3 }\). Coin \(B\) is weighted so that the probability of throwing a head is \(\frac { 1 } { 4 }\). Coin \(A\) is thrown twice and coin \(B\) is thrown once.
  1. Show that the probability of obtaining exactly 1 head and 2 tails is \(\frac { 13 } { 36 }\).
  2. Draw up the probability distribution table for the number of heads obtained.
  3. Find the expectation of the number of heads obtained.
CAIE S1 2014 June Q5
5 Find how many different numbers can be made from some or all of the digits of the number 1345789 if
  1. all seven digits are used, the odd digits are all together and no digits are repeated,
  2. the numbers made are even numbers between 3000 and 5000, and no digits are repeated,
  3. the numbers made are multiples of 5 which are less than 1000 , and digits can be repeated.
CAIE S1 2014 June Q6
6 The times taken by 57 athletes to run 100 metres are summarised in the following cumulative frequency table.
Time (seconds)\(< 10.0\)\(< 10.5\)\(< 11.0\)\(< 12.0\)\(< 12.5\)\(< 13.5\)
Cumulative frequency0410404957
  1. State how many athletes ran 100 metres in a time between 10.5 and 11.0 seconds.
  2. Draw a histogram on graph paper to represent the times taken by these athletes to run 100 metres.
  3. Calculate estimates of the mean and variance of the times taken by these athletes.
CAIE S1 2014 June Q7
7 The time Rafa spends on his homework each day in term-time has a normal distribution with mean 1.9 hours and standard deviation \(\sigma\) hours. On \(80 \%\) of these days he spends more than 1.35 hours on his homework.
  1. Find the value of \(\sigma\).
  2. Find the probability that, on a randomly chosen day in term-time, Rafa spends less than 2 hours on his homework.
  3. A random sample of 200 days in term-time is taken. Use an approximation to find the probability that the number of days on which Rafa spends more than 1.35 hours on his homework is between 163 and 173 inclusive.
CAIE S1 2014 June Q1
1 Some adults and some children each tried to estimate, without using a watch, the number of seconds that had elapsed in a fixed time-interval. Their estimates are shown below.
Adults:555867746361637156535478736462
Children:869589726184779281544368626783
  1. Draw a back-to-back stem-and-leaf diagram to represent the data.
  2. Make two comparisons between the estimates of the adults and the children.
CAIE S1 2014 June Q2
2 There is a probability of \(\frac { 1 } { 7 }\) that Wenjie goes out with her friends on any particular day. 252 days are chosen at random.
  1. Use a normal approximation to find the probability that the number of days on which Wenjie goes out with her friends is less than than 30 or more than 44.
  2. Give a reason why the use of a normal approximation is justified.
CAIE S1 2014 June Q3
3 A pet shop has 6 rabbits and 3 hamsters. 5 of these pets are chosen at random. The random variable \(X\) represents the number of hamsters chosen.
  1. Show that the probability that exactly 2 hamsters are chosen is \(\frac { 10 } { 21 }\).
  2. Draw up the probability distribution table for \(X\).
CAIE S1 2014 June Q4
4 The heights, \(x \mathrm {~cm}\), of a group of 28 people were measured. The mean height was found to be 172.6 cm and the standard deviation was found to be 4.58 cm . A person whose height was 161.8 cm left the group.
  1. Find the mean height of the remaining group of 27 people.
  2. Find \(\Sigma x ^ { 2 }\) for the original group of 28 people. Hence find the standard deviation of the heights of the remaining group of 27 people.
CAIE S1 2014 June Q5
5 When Moses makes a phone call, the amount of time that the call takes has a normal distribution with mean 6.5 minutes and standard deviation 1.76 minutes.
  1. \(90 \%\) of Moses's phone calls take longer than \(t\) minutes. Find the value of \(t\).
  2. Find the probability that, in a random sample of 9 phone calls made by Moses, more than 7 take a time which is within 1 standard deviation of the mean.
CAIE S1 2014 June Q6
6 Tom and Ben play a game repeatedly. The probability that Tom wins any game is 0.3 . Each game is won by either Tom or Ben. Tom and Ben stop playing when one of them (to be called the champion) has won two games.
  1. Find the probability that Ben becomes the champion after playing exactly 2 games.
  2. Find the probability that Ben becomes the champion.
  3. Given that Tom becomes the champion, find the probability that he won the 2nd game.
CAIE S1 2014 June Q7
7 Nine cards are numbered \(1,2,2,3,3,4,6,6,6\).
  1. All nine cards are placed in a line, making a 9-digit number. Find how many different 9-digit numbers can be made in this way
    (a) if the even digits are all together,
    (b) if the first and last digits are both odd.
  2. Three of the nine cards are chosen and placed in a line, making a 3-digit number. Find how many different numbers can be made in this way
    (a) if there are no repeated digits,
    (b) if the number is between 200 and 300 .
CAIE S1 2015 June Q1
1 The lengths, in metres, of cars in a city are normally distributed with mean \(\mu\) and standard deviation 0.714 . The probability that a randomly chosen car has a length more than 3.2 metres and less than \(\mu\) metres is 0.475 . Find \(\mu\).
CAIE S1 2015 June Q2
2 The table summarises the lengths in centimetres of 104 dragonflies.
Length \(( \mathrm { cm } )\)\(2.0 - 3.5\)\(3.5 - 4.5\)\(4.5 - 5.5\)\(5.5 - 7.0\)\(7.0 - 9.0\)
Frequency825283112
  1. State which class contains the upper quartile.
  2. Draw a histogram, on graph paper, to represent the data.
CAIE S1 2015 June Q3
3 Jason throws two fair dice, each with faces numbered 1 to 6 . Event \(A\) is 'one of the numbers obtained is divisible by 3 and the other number is not divisible by 3 '. Event \(B\) is 'the product of the two numbers obtained is even'.
  1. Determine whether events \(A\) and \(B\) are independent, showing your working.
  2. Are events \(A\) and \(B\) mutually exclusive? Justify your answer.
CAIE S1 2015 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{c0c7e038-805a-4237-a579-a6571b84f337-2_451_1530_1393_303} A survey is undertaken to investigate how many photos people take on a one-week holiday and also how many times they view past photos. For a randomly chosen person, the probability of taking fewer than 100 photos is \(x\). The probability that these people view past photos at least 3 times is 0.76 . For those who take at least 100 photos, the probability that they view past photos fewer than 3 times is 0.90 . This information is shown in the tree diagram. The probability that a randomly chosen person views past photos fewer than 3 times is 0.801 .
  1. Find \(x\).
  2. Given that a person views past photos at least 3 times, find the probability that this person takes at least 100 photos.