Questions — CAIE S1 (785 questions)

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CAIE S1 2009 November Q5
5
  1. Find how many numbers between 5000 and 6000 can be formed from the digits 1, 2, 3, 4, 5 and 6
    1. if no digits are repeated,
    2. if repeated digits are allowed.
  2. Find the number of ways of choosing a school team of 5 pupils from 6 boys and 8 girls
    1. if there are more girls than boys in the team,
    2. if three of the boys are cousins and are either all in the team or all not in the team.
CAIE S1 2009 November Q6
6 A box contains 4 pears and 7 oranges. Three fruits are taken out at random and eaten. Find the probability that
  1. 2 pears and 1 orange are eaten, in any order,
  2. the third fruit eaten is an orange,
  3. the first fruit eaten was a pear, given that the third fruit eaten is an orange. There are 121 similar boxes in a warehouse. One fruit is taken at random from each box.
  4. Using a suitable approximation, find the probability that fewer than 39 are pears.
CAIE S1 2009 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{6f677fc6-3ca2-4a0d-82a2-69a7cbb8574d-2_211_1169_267_488} Measurements of wind speed on a certain island were taken over a period of one year. A box-andwhisker plot of the data obtained is displayed above, and the values of the quartiles are as shown. It is suggested that wind speed can be modelled approximately by a normal distribution with mean \(\mu \mathrm { km } \mathrm { h } ^ { - 1 }\) and standard deviation \(\sigma \mathrm { km } \mathrm { h } ^ { - 1 }\).
  1. Estimate the value of \(\mu\).
  2. Estimate the value of \(\sigma\).
CAIE S1 2009 November Q2
2 Two unbiased tetrahedral dice each have four faces numbered \(1,2,3\) and 4. The two dice are thrown together and the sum of the numbers on the faces on which they land is noted. Find the expected number of occasions on which this sum is 7 or more when the dice are thrown together 200 times.
CAIE S1 2009 November Q3
3 Maria chooses toast for her breakfast with probability 0.85 . If she does not choose toast then she has a bread roll. If she chooses toast then the probability that she will have jam on it is 0.8 . If she has a bread roll then the probability that she will have jam on it is 0.4 .
  1. Draw a fully labelled tree diagram to show this information.
  2. Given that Maria did not have jam for breakfast, find the probability that she had toast.
CAIE S1 2009 November Q4
4
    1. Find how many different four-digit numbers can be made using only the digits 1, 3, 5 and 6 with no digit being repeated.
    2. Find how many different odd numbers greater than 500 can be made using some or all of the digits \(1,3,5\) and 6 with no digit being repeated.
  2. Six cards numbered 1,2,3,4,5,6 are arranged randomly in a line. Find the probability that the cards numbered 4 and 5 are not next to each other.
CAIE S1 2009 November Q5
5 In a particular discrete probability distribution the random variable \(X\) takes the value \(\frac { 120 } { r }\) with probability \(\frac { r } { 45 }\), where \(r\) takes all integer values from 1 to 9 inclusive.
  1. Show that \(\mathrm { P } ( X = 40 ) = \frac { 1 } { 15 }\).
  2. Construct the probability distribution table for \(X\).
  3. Which is the modal value of \(X\) ?
  4. Find the probability that \(X\) lies between 18 and 100 .
CAIE S1 2009 November Q6
6 The following table gives the marks, out of 75, in a pure mathematics examination taken by 234 students.
Marks\(1 - 20\)\(21 - 30\)\(31 - 40\)\(41 - 50\)\(51 - 60\)\(61 - 75\)
Frequency403456542921
  1. Draw a histogram on graph paper to represent these results.
  2. Calculate estimates of the mean mark and the standard deviation.
CAIE S1 2009 November Q7
7 The weights, \(X\) grams, of bars of soap are normally distributed with mean 125 grams and standard deviation 4.2 grams.
  1. Find the probability that a randomly chosen bar of soap weighs more than 128 grams.
  2. Find the value of \(k\) such that \(\mathrm { P } ( k < X < 128 ) = 0.7465\).
  3. Five bars of soap are chosen at random. Find the probability that more than two of the bars each weigh more than 128 grams.
CAIE S1 2010 November Q1
1 Anita made observations of the maximum temperature, \(t ^ { \circ } \mathrm { C }\), on 50 days. Her results are summarised by \(\Sigma t = 910\) and \(\Sigma ( t - \bar { t } ) ^ { 2 } = 876\), where \(\bar { t }\) denotes the mean of the 50 observations. Calculate \(\bar { t }\) and the standard deviation of the observations.
CAIE S1 2010 November Q2
2 On average, 2 apples out of 15 are classified as being underweight. Find the probability that in a random sample of 200 apples, the number of apples which are underweight is more than 21 and less than 35.
CAIE S1 2010 November Q3
3 The times taken by students to get up in the morning can be modelled by a normal distribution with mean 26.4 minutes and standard deviation 3.7 minutes.
  1. For a random sample of 350 students, find the number who would be expected to take longer than 20 minutes to get up in the morning.
  2. 'Very slow' students are students whose time to get up is more than 1.645 standard deviations above the mean. Find the probability that fewer than 3 students from a random sample of 8 students are 'very slow'.
CAIE S1 2010 November Q4
4 The weights in grams of a number of stones, measured correct to the nearest gram, are represented in the following table.
Weight (grams)\(1 - 10\)\(11 - 20\)\(21 - 25\)\(26 - 30\)\(31 - 50\)\(51 - 70\)
Frequency\(2 x\)\(4 x\)\(3 x\)\(5 x\)\(4 x\)\(x\)
A histogram is drawn with a scale of 1 cm to 1 unit on the vertical axis, which represents frequency density. The \(1 - 10\) rectangle has height 3 cm .
  1. Calculate the value of \(x\) and the height of the 51-70 rectangle.
  2. Calculate an estimate of the mean weight of the stones.
CAIE S1 2010 November Q5
5 Three friends, Rick, Brenda and Ali, go to a football match but forget to say which entrance to the ground they will meet at. There are four entrances, \(A , B , C\) and \(D\). Each friend chooses an entrance independently.
  • The probability that Rick chooses entrance \(A\) is \(\frac { 1 } { 3 }\). The probabilities that he chooses entrances \(B , C\) or \(D\) are all equal.
  • Brenda is equally likely to choose any of the four entrances.
  • The probability that Ali chooses entrance \(C\) is \(\frac { 2 } { 7 }\) and the probability that he chooses entrance \(D\) is \(\frac { 3 } { 5 }\). The probabilities that he chooses the other two entrances are equal.
    1. Find the probability that at least 2 friends will choose entrance \(B\).
    2. Find the probability that the three friends will all choose the same entrance.
CAIE S1 2010 November Q6
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
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
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
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
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
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
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
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
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
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
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.