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CAIE S1 2010 November Q6
9 marks Moderate -0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{fcf7b1c6-cc76-4c84-998c-9de6a7e9bb2d-3_163_618_260_765} Pegs are to be placed in the four holes shown, one in each hole. The pegs come in different colours and pegs of the same colour are identical. Calculate how many different arrangements of coloured pegs in the four holes can be made using
  1. 6 pegs, all of different colours,
  2. 4 pegs consisting of 2 blue pegs, 1 orange peg and 1 yellow peg. Beryl has 12 pegs consisting of 2 red, 2 blue, 2 green, 2 orange, 2 yellow and 2 black pegs. Calculate how many different arrangements of coloured pegs in the 4 holes Beryl can make using
  3. 4 different colours,
  4. 3 different colours,
  5. any of her 12 pegs.
CAIE S1 2010 November Q7
11 marks Standard +0.3
7 Sanket plays a game using a biased die which is twice as likely to land on an even number as on an odd number. The probabilities for the three even numbers are all equal and the probabilities for the three odd numbers are all equal.
  1. Find the probability of throwing an odd number with this die. Sanket throws the die once and calculates his score by the following method.
    • If the number thrown is 3 or less he multiplies the number thrown by 3 and adds 1 .
    • If the number thrown is more than 3 he multiplies the number thrown by 2 and subtracts 4 .
    The random variable \(X\) is Sanket's score.
  2. Show that \(\mathrm { P } ( X = 8 ) = \frac { 2 } { 9 }\). The table shows the probability distribution of \(X\).
    \(x\)467810
    \(\mathrm { P } ( X = x )\)\(\frac { 3 } { 9 }\)\(\frac { 1 } { 9 }\)\(\frac { 2 } { 9 }\)\(\frac { 2 } { 9 }\)\(\frac { 1 } { 9 }\)
  3. Given that \(\mathrm { E } ( X ) = \frac { 58 } { 9 }\), find \(\operatorname { Var } ( X )\). Sanket throws the die twice.
  4. Find the probability that the total of the scores on the two throws is 16 .
  5. Given that the total of the scores on the two throws is 16 , find the probability that the score on the first throw was 6 .
CAIE S1 2010 November Q1
3 marks Moderate -0.8
1 The discrete random variable \(X\) takes the values 1, 4, 5, 7 and 9 only. The probability distribution of \(X\) is shown in the table.
\(x\)14579
\(\mathrm { P } ( X = x )\)\(4 p\)\(5 p ^ { 2 }\)\(1.5 p\)\(2.5 p\)\(1.5 p\)
Find \(p\).
CAIE S1 2010 November Q2
6 marks Moderate -0.8
2 Esme noted the test marks, \(x\), of 16 people in a class. She found that \(\Sigma x = 824\) and that the standard deviation of \(x\) was 6.5.
  1. Calculate \(\Sigma ( x - 50 )\) and \(\Sigma ( x - 50 ) ^ { 2 }\).
  2. One person did the test later and her mark was 72. Calculate the new mean and standard deviation of the marks of all 17 people.
CAIE S1 2010 November Q3
6 marks Standard +0.8
3 A fair five-sided spinner has sides numbered 1,2,3,4,5. Raj spins the spinner and throws two fair dice. He calculates his score as follows.
  • If the spinner lands on an even-numbered side, Raj multiplies the two numbers showing on the dice to get his score.
  • If the spinner lands on an odd-numbered side, Raj adds the numbers showing on the dice to get his score.
Given that Raj's score is 12, find the probability that the spinner landed on an even-numbered side.
CAIE S1 2010 November Q4
7 marks Easy -1.8
4 The weights in kilograms of 11 bags of sugar and 7 bags of flour are as follows.
Sugar: 1.9611 .98312 .00812 .0141 .9681 .9941 .2 .0112 .0171 .9771 .9841 .989
Flour: \(\begin{array} { l l l l l l l } 1.945 & 1.962 & 1.949 & 1.977 & 1.964 & 1.941 & 1.953 \end{array}\)
  1. Represent this information on a back-to-back stem-and-leaf diagram with sugar on the left-hand side.
  2. Find the median and interquartile range of the weights of the bags of sugar.
CAIE S1 2010 November Q5
7 marks Moderate -0.3
5 The distance the Zotoc car can travel on 20 litres of fuel is normally distributed with mean 320 km and standard deviation 21.6 km . The distance the Ganmor car can travel on 20 litres of fuel is normally distributed with mean 350 km and standard deviation 7.5 km . Both cars are filled with 20 litres of fuel and are driven towards a place 367 km away.
  1. For each car, find the probability that it runs out of fuel before it has travelled 367 km .
  2. The probability that a Zotoc car can travel at least \(( 320 + d ) \mathrm { km }\) on 20 litres of fuel is 0.409 . Find the value of \(d\).
CAIE S1 2010 November Q6
10 marks Moderate -0.8
6
  1. State three conditions that must be satisfied for a situation to be modelled by a binomial distribution. On any day, there is a probability of 0.3 that Julie's train is late.
  2. Nine days are chosen at random. Find the probability that Julie's train is late on more than 7 days or fewer than 2 days.
  3. 90 days are chosen at random. Find the probability that Julie's train is late on more than 35 days or fewer than 27 days.
CAIE S1 2010 November Q7
11 marks Standard +0.3
7 A committee of 6 people, which must contain at least 4 men and at least 1 woman, is to be chosen from 10 men and 9 women.
  1. Find the number of possible committees that can be chosen.
  2. Find the probability that one particular man, Albert, and one particular woman, Tracey, are both on the committee.
  3. Find the number of possible committees that include either Albert or Tracey but not both.
  4. The committee that is chosen consists of 4 men and 2 women. They queue up randomly in a line for refreshments. Find the probability that the women are not next to each other in the queue.
CAIE S1 2012 November Q1
5 marks Moderate -0.8
1 Fabio drinks coffee each morning. He chooses Americano, Cappucino or Latte with probabilities 0.5, 0.3 and 0.2 respectively. If he chooses Americano he either drinks it immediately with probability 0.8 , or leaves it to drink later. If he chooses Cappucino he either drinks it immediately with probability 0.6 , or leaves it to drink later. If he chooses Latte he either drinks it immediately with probability 0.1 , or leaves it to drink later.
  1. Find the probability that Fabio chooses Americano and leaves it to drink later.
  2. Fabio drinks his coffee immediately. Find the probability that he chose Latte.
CAIE S1 2012 November Q2
7 marks Standard +0.3
2 The random variable \(X\) is the daily profit, in thousands of dollars, made by a company. \(X\) is normally distributed with mean 6.4 and standard deviation 5.2.
  1. Find the probability that, on a randomly chosen day, the company makes a profit between \(\\) 10000\( and \)\\( 12000\).
  2. Find the probability that the company makes a loss on exactly 1 of the next 4 consecutive days.
CAIE S1 2012 November Q3
8 marks Easy -1.2
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
9 marks Standard +0.8
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
10 marks Standard +0.3
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
11 marks Standard +0.3
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
3 marks Moderate -0.3
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
6 marks Standard +0.8
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
8 marks Easy -1.8
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
9 marks Moderate -0.8
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
12 marks Standard +0.3
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
12 marks Standard +0.3
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
3 marks Easy -1.8
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 Moderate -0.3
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
5 marks Moderate -0.8
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
7 marks Moderate -0.8
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.