Questions — CAIE (7646 questions)

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CAIE S1 2016 June Q3
6 marks Moderate -0.8
3 Two ordinary fair dice are thrown. The resulting score is found as follows.
  • If the two dice show different numbers, the score is the smaller of the two numbers.
  • If the two dice show equal numbers, the score is 0 .
    1. Draw up the probability distribution table for the score.
    2. Calculate the expected score.
CAIE S1 2016 June Q4
6 marks Moderate -0.8
4 The monthly rental prices, \(\\) x$, for 9 apartments in a certain city are listed and are summarised as follows. $$\Sigma ( x - c ) = 1845 \quad \Sigma ( x - c ) ^ { 2 } = 477450$$ The mean monthly rental price is \(\\) 2205$.
  1. Find the value of the constant \(c\).
  2. Find the variance of these values of \(x\).
  3. Another apartment is added to the list. The mean monthly rental price is now \(\\) 2120.50$. Find the rental price of this additional apartment.
CAIE S1 2016 June Q5
8 marks Standard +0.3
5 The heights of school desks have a normal distribution with mean 69 cm and standard deviation \(\sigma \mathrm { cm }\). It is known that 15.5\% of these desks have a height greater than 70 cm .
  1. Find the value of \(\sigma\). When Jodu sits at a desk, his knees are at a height of 58 cm above the floor. A desk is comfortable for Jodu if his knees are at least 9 cm below the top of the desk. Jodu's school has 300 desks.
  2. Calculate an estimate of the number of these desks that are comfortable for Jodu.
CAIE S1 2016 June Q6
9 marks Moderate -0.8
6 Find the number of ways all 9 letters of the word EVERGREEN can be arranged if
  1. there are no restrictions,
  2. the first letter is R and the last letter is G ,
  3. the Es are all together. Three letters from the 9 letters of the word EVERGREEN are selected.
  4. Find the number of selections which contain no Es and exactly 1 R .
  5. Find the number of selections which contain no Es.
CAIE S1 2016 June Q7
11 marks Standard +0.3
7 Passengers are travelling to Picton by minibus. The probability that each passenger carries a backpack is 0.65 , independently of other passengers. Each minibus has seats for 12 passengers.
  1. Find the probability that, in a full minibus travelling to Picton, between 8 passengers and 10 passengers inclusive carry a backpack.
  2. Passengers get on to an empty minibus. Find the probability that the fourth passenger who gets on to the minibus will be the first to be carrying a backpack.
  3. Find the probability that, of a random sample of 250 full minibuses travelling to Picton, more than 54 will contain exactly 7 passengers carrying backpacks.
CAIE S1 2017 June Q1
4 marks Moderate -0.8
1 Kadijat noted the weights, \(x\) grams, of 30 chocolate buns. Her results are summarised by $$\Sigma ( x - k ) = 315 , \quad \Sigma ( x - k ) ^ { 2 } = 4022$$ where \(k\) is a constant. The mean weight of the buns is 50.5 grams.
  1. Find the value of \(k\).
  2. Find the standard deviation of \(x\).
CAIE S1 2017 June Q2
5 marks Moderate -0.5
2 Ashfaq throws two fair dice and notes the numbers obtained. \(R\) is the event 'The product of the two numbers is 12 '. \(T\) is the event 'One of the numbers is odd and one of the numbers is even'. By finding appropriate probabilities, determine whether events \(R\) and \(T\) are independent.
CAIE S1 2017 June Q3
6 marks Standard +0.3
3 Redbury United soccer team play a match every week. Each match can be won, drawn or lost. At the beginning of the soccer season the probability that Redbury United win their first match is \(\frac { 3 } { 5 }\), with equal probabilities of losing or drawing. If they win the first match, the probability that they win the second match is \(\frac { 7 } { 10 }\) and the probability that they lose the second match is \(\frac { 1 } { 10 }\). If they draw the first match they are equally likely to win, draw or lose the second match. If they lose the first match, the probability that they win the second match is \(\frac { 3 } { 10 }\) and the probability that they draw the second match is \(\frac { 1 } { 20 }\).
  1. Draw a fully labelled tree diagram to represent the first two matches played by Redbury United in the soccer season.
  2. Given that Redbury United win the second match, find the probability that they lose the first match.
CAIE S1 2017 June Q4
6 marks Easy -1.2
4 The times taken, \(t\) seconds, by 1140 people to solve a puzzle are summarised in the table.
Time \(( t\) seconds \()\)\(0 \leqslant t < 20\)\(20 \leqslant t < 40\)\(40 \leqslant t < 60\)\(60 \leqslant t < 100\)\(100 \leqslant t < 140\)
Number of people320280220220100
  1. On the grid, draw a histogram to illustrate this information. \includegraphics[max width=\textwidth, alt={}, center]{7652f36c-59b5-4fcd-b17b-d796dc82aec0-05_812_1406_804_411}
  2. Calculate an estimate of the mean of \(t\).
CAIE S1 2017 June Q5
7 marks Moderate -0.3
5 Eggs are sold in boxes of 20. Cracked eggs occur independently and the mean number of cracked eggs in a box is 1.4 .
  1. Calculate the probability that a randomly chosen box contains exactly 2 cracked eggs.
  2. Calculate the probability that a randomly chosen box contains at least 1 cracked egg.
  3. A shop sells \(n\) of these boxes of eggs. Find the smallest value of \(n\) such that the probability of there being at least 1 cracked egg in each box sold is less than 0.01 .
CAIE S1 2017 June Q6
11 marks Standard +0.3
6
  1. The random variable \(X\) has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). You are given that \(\sigma = 0.25 \mu\) and \(\mathrm { P } ( X < 6.8 ) = 0.75\).
    1. Find the value of \(\mu\).
    2. Find \(\mathrm { P } ( X < 4.7 )\).
  2. The lengths of metal rods have a normal distribution with mean 16 cm and standard deviation 0.2 cm . Rods which are shorter than 15.75 cm or longer than 16.25 cm are not usable. Find the expected number of usable rods in a batch of 1000 rods.
CAIE S1 2017 June Q7
11 marks Standard +0.3
7
  1. Eight children of different ages stand in a random order in a line. Find the number of different ways this can be done if none of the three youngest children stand next to each other.
  2. David chooses 5 chocolates from 6 different dark chocolates, 4 different white chocolates and 1 milk chocolate. He must choose at least one of each type. Find the number of different selections he can make.
  3. A password for Chelsea's computer consists of 4 characters in a particular order. The characters are chosen from the following.
    The password must include at least one capital letter, at least one digit and at least one symbol. No character can be repeated. Find the number of different passwords that Chelsea can make.
CAIE S1 2017 June Q1
5 marks Easy -1.2
1 Rani and Diksha go shopping for clothes.
  1. Rani buys 4 identical vests, 3 identical sweaters and 1 coat. Each vest costs \(\\) 5.50\( and the coat costs \)\\( 90\). The mean cost of Rani's 8 items is \(\\) 29\(. Find the cost of a sweater.
  2. Diksha buys 1 hat and 4 identical shirts. The mean cost of Diksha's 5 items is \)\\( 26\) and the standard deviation is \(\\) 0\(. Explain how you can tell that Diksha spends \)\\( 104\) on shirts.
CAIE S1 2017 June Q2
6 marks Easy -1.8
2 Anabel measured the lengths, in centimetres, of 200 caterpillars. Her results are illustrated in the cumulative frequency graph below. \includegraphics[max width=\textwidth, alt={}, center]{184a04ac-4396-4a0f-8fa8-ab11a4b6df39-03_1173_1195_356_466}
  1. Estimate the median and the interquartile range of the lengths.
  2. Estimate how many caterpillars had a length of between 2 and 3.5 cm .
  3. 6\% of caterpillars were of length \(l\) centimetres or more. Estimate \(l\).
CAIE S1 2017 June Q3
6 marks Moderate -0.8
3 In a probability distribution the random variable \(X\) takes the value \(x\) with probability \(k x ^ { 2 }\), where \(k\) is a constant and \(x\) takes values \(- 2 , - 1,2,4\) only.
  1. Show that \(\mathrm { P } ( X = - 2 )\) has the same value as \(\mathrm { P } ( X = 2 )\).
  2. Draw up the probability distribution table for \(X\), in terms of \(k\), and find the value of \(k\).
  3. Find \(\mathrm { E } ( X )\).
CAIE S1 2017 June Q4
6 marks Standard +0.3
4 Two identical biased triangular spinners with sides marked 1,2 and 3 are spun. For each spinner, the probabilities of landing on the sides marked 1,2 and 3 are \(p , q\) and \(r\) respectively. The score is the sum of the numbers on the sides on which the spinners land. You are given that \(\mathrm { P } (\) score is \(6 ) = \frac { 1 } { 36 }\) and \(\mathrm { P } (\) score is \(5 ) = \frac { 1 } { 9 }\). Find the values of \(p , q\) and \(r\).
CAIE S1 2017 June Q5
9 marks Standard +0.3
5 The lengths of videos of a certain popular song have a normal distribution with mean 3.9 minutes. \(18 \%\) of these videos last for longer than 4.2 minutes.
  1. Find the standard deviation of the lengths of these videos.
  2. Find the probability that the length of a randomly chosen video differs from the mean by less than half a minute.
    The lengths of videos of another popular song have a normal distribution with the same mean of 3.9 minutes but the standard deviation is twice the standard deviation in part (i). The probability that the length of a randomly chosen video of this song differs from the mean by less than half a minute is denoted by \(p\).
  3. Without any further calculation, determine whether \(p\) is more than, equal to, or less than your answer to part (ii). You must explain your reasoning.
CAIE S1 2017 June Q6
9 marks Standard +0.8
6 A library contains 4 identical copies of book \(A , 2\) identical copies of book \(B\) and 5 identical copies of book \(C\). These 11 books are arranged on a shelf in the library.
  1. Calculate the number of different arrangements if the end books are either both book \(A\) or both book \(B\).
  2. Calculate the number of different arrangements if all the books \(A\) are next to each other and none of the books \(B\) are next to each other.
CAIE S1 2017 June Q7
9 marks Moderate -0.8
7 During the school holidays, each day Khalid either rides on his bicycle with probability 0.6 , or on his skateboard with probability 0.4 . Khalid does not ride on both on the same day. If he rides on his bicycle then the probability that he hurts himself is 0.05 . If he rides on his skateboard the probability that he hurts himself is 0.75 .
  1. Find the probability that Khalid hurts himself on any particular day.
  2. Given that Khalid hurts himself on a particular day, find the probability that he is riding on his skateboard.
  3. There are 45 days of school holidays. Show that the variance of the number of days Khalid rides on his skateboard is the same as the variance of the number of days that Khalid rides on his bicycle.
  4. Find the probability that Khalid rides on his skateboard on at least 2 of 10 randomly chosen days in the school holidays.
CAIE S1 2017 June Q1
4 marks Moderate -0.3
1 A biased die has faces numbered 1 to 6 . The probabilities of the die landing on 1,3 or 5 are each equal to 0.1 . The probabilities of the die landing on 2 or 4 are each equal to 0.2 . The die is thrown twice. Find the probability that the sum of the numbers it lands on is 9 .
CAIE S1 2017 June Q2
5 marks Standard +0.3
2 The probability that George goes swimming on any day is \(\frac { 1 } { 3 }\). Use an approximation to calculate the probability that in 270 days George goes swimming at least 100 times.
CAIE S1 2017 June Q3
5 marks Moderate -0.8
3 A shop sells two makes of coffee, Café Premium and Café Standard. Both coffees come in two sizes, large jars and small jars. Of the jars on sale, \(65 \%\) are Café Premium and \(35 \%\) are Café Standard. Of the Café Premium, 40\% of the jars are large and of the Café Standard, 25\% of the jars are large. A jar is chosen at random.
  1. Find the probability that the jar is small.
  2. Find the probability that the jar is Café Standard given that it is large.
CAIE S1 2017 June Q4
6 marks Standard +0.3
4
  1. The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\), where \(\mu = 1.5 \sigma\). A random value of \(X\) is chosen. Find the probability that this value of \(X\) is greater than 0 .
  2. The life of a particular type of torch battery is normally distributed with mean 120 hours and standard deviation \(s\) hours. It is known that \(87.5 \%\) of these batteries last longer than 70 hours. Find the value of \(s\).
CAIE S1 2017 June Q5
8 marks Moderate -0.8
5 Hebe attempts a crossword puzzle every day. The number of puzzles she completes in a week (7 days) is denoted by \(X\).
  1. State two conditions that are required for \(X\) to have a binomial distribution.
    On average, Hebe completes 7 out of 10 of these puzzles.
  2. Use a binomial distribution to find the probability that Hebe completes at least 5 puzzles in a week.
  3. Use a binomial distribution to find the probability that, over the next 10 weeks, Hebe completes 4 or fewer puzzles in exactly 3 of the 10 weeks.
CAIE S1 2017 June Q6
11 marks Moderate -0.8
6
  1. Find how many numbers between 3000 and 5000 can be formed from the digits \(1,2,3,4\) and 5,
    1. if digits are not repeated,
    2. if digits can be repeated and the number formed is odd.
  2. A box of 20 biscuits contains 4 different chocolate biscuits, 2 different oatmeal biscuits and 14 different ginger biscuits. 6 biscuits are selected from the box at random.
    1. Find the number of different selections that include the 2 oatmeal biscuits.
    2. Find the probability that fewer than 3 chocolate biscuits are selected.