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mathematics
cambridge international as & a level mathematics probability & statistics
Cambridge International AS & A Level Mathematics Probability & Statistics 1 Coursebook 1st Edition Dean Chalmers, Julian Gilbey - Solutions
The letters A, B, B, B, C, D,D and E are written onto eight cards and placed in a bag. Find the probability that the letter on a randomly selected card is:a. A vowel or in the word DOMAINb. A consonant or in the word DOUBLE.
The probabilities that a team wins, draws or loses any particular game are 0.6, 0.1 and 0.3, respectively.a. Find the probability that the team wins at least one of its next two games.b. If 2 points are awarded for a win, 1 point for a draw and 0 points for a loss, find the probability that the
In a class of 28 children, 19 attend drama classes, 13 attend singing lessons, and six attend both drama classes and singing lessons. One child is chosen at random from the class.Event D is ‘a child who attends drama classes is chosen’.Event S is ‘a child who attends singing lessons is
The quiz marks of 40 students are represented in the following bar chart.Two students are selected at random from the group. Find the probability that the second student:a. Scored more than 5, given that the first student did not score more than 5b. Scored more than 7, given that the first student
Two children are randomly selected from a group of five boys and seven girls. Determine which is more likely to be selected:a. Two boys or two girls?b. The two youngest girls or the two oldest boys?
When they are switched on, certain small devices independently produce outputs of 1, 2 or 3 volts with respective probabilities of 0.3, 0.6 and 0.1. Find the probability that three of these devices produce an output with a sum of 5 or 6 volts.
In a group of 25 boys, nine are members of the chess club (C), eight are members of the debating club (D) and 10 are members of neither of these clubs. This information is shown in the Venn diagram.a. Find the values of a, b and c.b. Find the probability that a randomly selected boy is:i. A member
On any particular day, there is a 30% chance of snow in Slushly. Find the probability that it snows there on:a. None of the next 3 days b. Exactly one of the next 3 days.
Each child in a group of 80 was asked whether they regularly read (R) or regularly watch a movie (M). The results are given in the Venn diagram opposite. One child is selected at random from the group. Event R is ‘a child who regularly reads is selected’ and event M is ‘a child who regularly
The histogram shown represents the times taken, in minutes, for 115 men to complete a task.Two men are selected at random from the group. Find the probability that the:a. First man took less than 1 minute, given that he took less than 3 minutesb. Second man took less than 6 minutes, given that the
A boy has five different pairs of shoes mixed up under his bed. Find the probability that when he selects two shoes at random they can be worn as a matching pair.
One hundred people are attending a conference. The following Venn diagram shows how many are male (M), have brown eyes (BE) and are right-handed (RH).a. Given that there are 43 males with brown eyes, 42 right-handed males and 46 right-handed people with brown eyes, copy and complete the Venn
A set of data values is 8,13,17,18, 24, 32, 34 and 38. Find the probability that a randomly selected value is more than one standard deviation from the mean.
Forty girls were asked to name the capital of Cuba and of Hungary; 19 knew the capital of Cuba, 20 knew the capital of Hungary and seven knew both.a. Draw a Venn diagram showing the number of girls who knew each of these capitals.b. Find the probability that a randomly selected girl knew:i. The
Fatima will enter three sporting events at the weekend. Her chances of winning each of them are shown in the following table.a. Assuming that the three events are independent, find the probability that Fatima wins:i. The shot put and discus ii. The shot put and discus only iii. Exactly
Two fair 4-sided dice, both with faces marked 1, 2, 3 and 4, are rolled.Event A is ‘the sum of the numbers obtained is a prime number’.Event B is ‘the product of the numbers obtained is an even number’.a. Find, in simplest form, the value of P(A), of P(B) and of P(A B).b. Determine, with
At an insurance company, 60% of the staff are male (M) and 70% work full-time (FT). The following Venn diagram shows this and one other piece of information.a. What information is given by the value 0.10 in the Venn diagram?b. Find the value of a, of b and of c.c. An employee is randomly selected.
A bag contains five 4cm nails, six 7cm nails and nine 10cm nails. Find the probability that two randomly selected nails: a. Have a total length of 14cmb. Are both 7cm long, given that they have a total length of 14cm.
A student travels to college by either of two routes, A or B. The probability that they use route A is 0.3, and the probability that they are passed by a bus on their way to college on any particular day is 0.034.They are twice as likely to be passed by a bus when they use route B as when they use
One student is randomly selected from a school that has 837 boys. The probability that a girl is selected is 4/7. Find the probability that a particular boy is selected.
In a survey on pet ownership, 36% of the participants own a cat, 20% own a hamster but not a cat, and 8% own a hamster and a cat. What percentage of the participants owns neither a hamster nor a cat?
Two ordinary fair dice are rolled.Event X is ‘the product of the two numbers obtained is odd’.Event Y is ‘the sum of the two numbers obtained is a multiple of 3’.a. Determine, giving reasons for your answer, whether X and Y are independent.b. Are events X and Y mutually exclusive? Justify
Two fair triangular spinners, both with sides marked 1, 2 and 3, are spun. Given that the sum of the two numbers spun is even, find the probability that the two numbers are the same.
Yvonne and Novac play two games of tennis every Saturday. Yvonne has a 65% chance of winning the first game and, if she wins it, her chances of winning the second game increase to 70%. However, if she loses the first game, then her chances of winning the second game decrease to 55%. Find the
Two ordinary fair dice are rolled. If the first shows a number less than 3, then the score is the mean of the numbers obtained; otherwise the score is equal to half the absolute (non-negative) difference between the numbers obtained. Find the probability that the score is: a. Positiveb.
A garage repaired 132 vehicles last month. The number of vehicles that required electrical (E), mechanical (M) and bodywork (B) repairs are given in the diagram opposite.Find the probability that a randomly selected vehicle required:a. Mechanical or bodywork repairsb. No bodywork repairsc. Exactly
A fair 8-sided die has faces marked 1, 2, 3, 4, 5, 6, 7 and 8. The score when the die is rolled is the number on the face that the die lands on. The die is rolled twice.Event V is ‘one of the scores is exactly 4 less than the other score’.Event W is ‘the product of the scores is less than
Letters and packages can take up to 2 days to be delivered by Speedipost couriers. The following table shows the percentage of items delivered at certain times after sending.a. Is there any truth in the statement ‘If you post 10 letters on Monday then only nine of them will be delivered before
Two ordinary fair dice are rolled and the two numbers rolled are added together to give the score. Given that a player’s score is greater than 6, find the probability that it is not greater than 8.
a. Given P(X ∩ Y ) = 0.13 and P(X ) = 0.65, find P(Y | X ).b. Given P(M ∩ N) = 0.27 and P(N |M) = 0.81, find P(M).c. Given P(V ∩ W) = 0.35 and P(W') = 0.60, find P(V |W).
The 100 students at a technical college must study at least one subject from Pure Mathematics (P), Statistics (S) and Mechanics (M). The numbers studying these subjects are given in the diagram opposite.a. Who does the number 17 in the diagram refer to?b. Find the probability that a randomly
The following histogram represents the results of a national survey on bus departure delay times.Two buses are selected at random. Calculate an estimate of the probability that:a. Both departures were delayed by less than 4 minutesb. at least one of the buses departed more than 7 minutes late.
Two hundred children are categorised by gender and by whether or not they own a bicycle. Of the 108 males, 60 own a bicycle, and altogether 90 children do not own a bicycle.a. Tabulate these data.b. Determine, giving reasons for your answer, whether ownership of a bicycle is independent of gender
The circular archery target shown, on which 1, 2, 3 or 5 points can be scored, is divided into four parts of unequal area by concentric circles. The radii of the circles are 3cm, 9cm,15cm and 30cm. You may assume that a randomly fired arrow pierces just one of the four areas and is equally likely
When a customer at a furniture store makes a purchase, there is a 15% chance that they purchase a bed. Given that 4.2% of all customers at the store purchase a bed, find the probability that a customer does not make a purchase at the store.
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.i. Draw a fully
Events X and Y are such that P(X ) = 0.5, P(Y ) = 0.6 and P(X ∩ Y ) = 0.2.a. State, giving a reason, whether events X and Y are mutually exclusive.b Using a Venn diagram, or otherwise, find P(X ∪ Y ).c. Find the probability that X or Y, but not both, occurs
Praveen wants to speak on the telephone to his friend. When his friend’s phone rings, he answers it with constant probability 0.6. If Praveen’s friend doesn’t answer his phone, Praveen will call later, but he will only try four times altogether. Find the probability that Praveen speaks with
At an election, it was found that people voted for Party X independently of their income group. The following table shows that 12400 people from three income groups voted altogether, and that 7440 of them voted for Party X.Find the value of a, of b and of c. LOW Medium High Cotals Ca a Totais 7440
A number between 10 and 100 inclusive is selected at random. Find the probability that the number is a multiple of 5, given that none of its digits is a 5.
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.One of these countries is chosen at random.i. Find
A, B and C are events where P(A) = 0.3, P(B) = 0.4, P(C) = 0.3, P(A ∩ B) = 0.12, P(A ∩ C) = 0 and P(B ∩ C) = 0.1.a. State which pair of events from A, B and C is mutually exclusive.b. Using a Venn diagram, or otherwise, find P[(A ∪ B ∪ C)'], which is the probability that neither A nor B
Each morning, Ruma randomly selects and buys one of the four newspapers available at her local shop. Find the probability that she buys:a. The same newspaper on two consecutive morningsb. Three different newspapers on three consecutive mornings.
The speed limit at a motorway junction is 120km/h. Information about the speeds and directions in which 207 vehicles were being driven are shown in the following table.Providing evidence to support your answer, determine which vehicles’ speeds were independent of their direction of travel. North
Three of Mr Jumbillo’s seven children, who include one set of twins, are selected at random.a. Calculate the probability that exactly one of the twins is selected.b. Given that exactly three of the children are girls, find the probability that the selection of three children contains more girls
Three boxes, A, B and C, each contain orange balls and blue balls, as shown.a. A girl selects a ball at random from a randomly selected box. Given that she selects a blue ball, find the probability that it is from box C.b. A boy randomly selects one ball from each box. Given that he selects exactly
The diagram opposite shows a 30cm square board with two rectangular cards attached. The 15cm by 20cm card covers one-quarter of the 8cm by 12cm card.A dart is randomly thrown at the board, so that it sticks within its perimeter. Use areas to calculate the probability that the dart pierces:a. Both
A coin is biased such that the probability that three tosses all result in heads is 125/512. Find the probability of obtaining no heads with three tosses of the coin.
Anya calls Zara once each evening before she goes to bed. She calls Zara’s mobile phone with probability 0.8 or her landline. The probability that Zara answers her mobile phone is 0.74, and the probability that she answers her landline is y.
A and B are independent events. If P(A) = 0.45 and P(B)=0.64, find P[(A ∪ B)'].
Given that P(A) = 0.4, P(B) = 0.7 and that P(A ∪ B) = 0.8, find:a. P(A ∪ B')b. P(A' ∩ B)
In a group of five men and four women, there are three pairs of male and female business partners and three teachers, where no teacher is in a business partnership. One man and one woman are selected at random. Find the probability that they are:a. Both teachersb. In a business partnership with
Every Friday, Arif offers to take his sons to the beach or to the park. The sons refuse an offer to the beach with probability 0.65 and accept an offer to the park with probability 0.85. The probability that Arif offers to take them to the beach is x. This
Two ordinary fair dice are rolled.Event A is ‘the sum of the numbers rolled is 2, 3 or 4’.Event B is ‘the absolute difference between the numbers rolled is 2, 3 or 4’.a. When the dice are rolled, they show a 1 and a 3. Explain why this result shows that events A and B are not mutually
Each of 27 tourists was asked which of the countries Angola (A), Burundi (B) and Cameroon (C) they had visited. Of the group, 15 had visited Angola; 8 had visited Burundi; 12 had visited Cameroon; 2 had visited all three countries; and 21 had visited only one. Of those who had visited Angola, 4 had
A biased die in the shape of a pyramid has five faces marked 1, 2, 3, 4 and 5. The possible scores are 1, 2, 3, 4 and 5 and p(x) = k – x/25, where k is a constant.a. Find, in terms of k, the probability of scoring:i. 5ii. Less than 3.b. The die is rolled three times and the scores are added
Two children are selected from a group in which there are 10 more boys than girls. Given that there are 756 equiprobable ordered selections that can occur, find the probability that two boys or two girls are selected.
A and B are independent events, where P(A ∩ B') = 0.14, P(A' ∪ B) = 0.39 andP(A ∩ B) < 0.25. Use a Venn diagram, or otherwise, to find P[(A ∪ B)'].
A game board is shown in the diagram. Players take turns to roll an ordinary fair die, then move their counters forward from ‘start’ a number of squares equal to the number rolled with the die.If a player’s counter ends its move on a coloured square, then it is moved back to the start.a. Find
There is a 43% chance that Riya meets her friend Jasmine when she travels to work. Given that Riya walks to work and does not meet Jasmine 30% of the time, and that she travels to work by a different method and meets Jasmine 25% of the time, find the probability that Riya walks to work on any
In a survey, adults were asked to answer yes or no to the question ‘Do you regularly watch the evening TV news?’ Some of the results from the survey are detailed in the Venn diagram opposite.One adult is selected at random and it is found that the events ‘a female is selected’ and ‘a
Bookings made at a hotel include a room plus any meal combination of breakfast (B), lunch (L) and supper (S). The Venn diagram opposite shows the number of each type of booking made by 71 guests on Friday.a. A guest who has not booked all three meals is selected at random. Find the probability that
Aaliyah buys a randomly selected magazine that contains a crossword puzzle on five randomly chosen days of each week. On 84% of the occasions that she buys a magazine, she attempts its crossword, which she manages to complete 60% of the time. Find the probability that, on any particular day,
Three strangers meet on a train. Assuming that a person is equally likely to be born in any of the 12 months of the year, find the probability that at least two of these three people were born in the same month of the year.
A box contains three black and four white chess pieces. Find the probability that a random selection of five chess pieces, taken one at a time without replacement, contains exactly two black pieces which are selected one immediately after the other.
The following table shows the numbers of IGCSE (I) and A Level (A) examinations passed by a group of university students.a. For a student selected at random, find:i. P(I + A = 11 | A < 4)ii. P(I − A > 5 | I + A > 10)b. Six students who all have at least three A Level passes are selected
Five investors are repaid, each with their initial investment increased by p% plus a fixed ‘thank you’ bonus of $q. The woman who invested $20000 is repaid double her investment and the man who invested $7500 is repaid triple his investment. Find the total amount that the five people invested,
Homes in a certain neighbourhood have recently sold for $220000, $242000, $236000 and $3500000. A potential buyer wants to know the average selling price in the neighbourhood. Which of the mean, median or mode would be more helpful? Explain your answer.
The numbers of items returned to the electrical department of a store on each of 100 consecutive days are given in the following table.a. Write down the median.b. Is the mode a good value to use as the average in this case? Give a reason for your answer.c. Find the value of p, given that a
The following table shows the hourly rates of pay, in dollars, of a company’s employees.a. Is the mean a good average to use here? Give a reason for your answer.b. Find the mean rate of pay for the majority of the employees. Hourly rate 6 7 109 No. emploe 8. 11 17 1 bria
A dataset of 20 values is denoted by x where Σ(x −1) = 58. Another dataset of 30 values is denoted by y where Σ(y − 2) = 36. Find the mean of the 50 values of x and y.
One of the units used to measure pressure is pounds per square inch (psi). The mean pressure in the four tyres of a particular vehicle is denoted by x̅psi. Given that 1 pound is approximately equal to 0.4536kg and that 1metre is approximately equal to 39.37 inches, express the sum of the pressures
A study was carried out on 60 electronic items to find the currents, x amperes, that could be safely passed through them at a fixed voltage before they overheat. The results are given in the two tables below.a. Find the value of p, of q and of r.b. Cumulative frequency graphs are drawn to
As part of a data collection exercise, members of a certain school year group were asked how long they spent on their Mathematics homework during one particular week. The times are given to the nearest 0.1 hour. The results are displayed in the following table.i. Draw, on graph paper, a histogram
A train makes a non-stop journey from one city to another and back again each day. Over a period of 30 days, the mean number of passengers per journey is exactly 61.5. Exact one-way ticket prices paid by these passengers are given by percentage in the following table.a. Calculate the total revenue
Students investigated the prices in dollars ($) of 1 litre bottles of a certain drink at 24 shops in a town and at 16 shops in surrounding villages. Denoting the town prices by t and the village prices by v, the students’ data are summarised by the totals Σ(t −1.1) = 1.44 and Σ(v −1.2)
The lengths of extra-time, t minutes, played in the first and second halves of 100 football matches are summarised in the following table.a. Explain how you know that the median extra-time played in the second halves is greater than in the first halves.b. The first-half median is exactly 100
For 150 values of x, it is given that Σ(x −1) + Σ(x − 4) = 4170. Find x̅.
The heights, in centimetres, of 54 children are represented in the following diagram. The children are split into two equal-sized groups: a ‘tall half’ and a ‘short half’.Calculate an estimate of the difference between the mean heights of these two groups of children. 140 144 150 156 159
Eighty candidates took an examination in Astronomy, for which no candidate scored more than 80%. The examiners suggest that five grades, A, B, C, D and E, should be awarded to these candidates, using upper grade boundaries 64, 50, 36 and 26 for grades B, C, D and E, respectively. In this case,
On Monday, a teacher asked eight students to write down a number, which is denoted by x. On Tuesday, when one of these students was absent, they asked them to add 1 to yesterday’s number and write it down.Find the number written down on Monday by the student who was absent on Tuesday, given that
The following table summarises the number of tomatoes produced by the plants in the plots on a farm.a. Calculate an estimate of the mean number of tomatoes produced by these plots.b. The tomatoes are weighed accurately and their mean mass is found to be 156.50 grams. At market they are sold for
The values of x shown in the following table are to be represented in a bar chart.a. i. Sketch a curve that shows the shape of the data.ii. Find the mode, mean and the median of x.b. The two smallest values of x (i.e. 5 and 5) are changed to 21 and 31. Investigate the effect that this has on the
A delivery of 150 boxes, each containing 20 items, is made to a retailer. The numbers of damaged items in the boxes are shown in the following table.a. Find the mode, the mean and the median number of damaged items.b. Which of the three measures of central tendency would be the most appropriate to
Twenty boys and girls were each asked how many aunts and uncles they have. The entry 4 / 5 in the following table, for example, shows that 4 boys and 5 girls each have 3 aunts and 2 uncles.a. Find the mean number of uncles that the boys have.b. For the boys and girls together, calculate the mean
A histogram is drawn to illustrate a set of continuous data whose mean and median are equal. Make sketches of the different types of curve that could be drawn to represent the shape of the histogram.
The monthly salaries, w dollars, of 10 women are such that Σ(w − 3000) = −200.The monthly salaries, m dollars, of 20 men are such that Σ(m − 4000) = 120.a. Find the difference between the mean monthly salary of the women and the mean monthly salary of the men.b. Find the mean monthly
A calculated estimate of the mean capacity of 120 refrigerators stored at a warehouse is 348 litres. The capacities are given in the following table.A delivery of n new refrigerators, all with capacities between 200 and 320 litres arrives at the warehouse. This causes the mean capacity to decrease
a. Without drawing an accurate histogram, describe the shape of the set of marks. What does the shape suggest about the values of the mean, the median and the mode?b. Information is provided about the marks in examinations in two other subjects: Chemistry: mode > median > mean Physics: mean
For 90 values of x and 64 values of y, it is given that Σ(x −1) = 72.9 and Σ(y +1) = 201.6. Find the mean value of all the values of x and y combined.
A carpet fitter is employed to fit carpet in each of the 72 guest bedrooms at a new hotel. The following table shows how many rooms were completed during the first 10 days of work.Based on these figures, estimate how many more days it will take to finish the job. What assumptions are you making in
In the figure opposite, a square of side 8cm is joined edge-to-edge to a semicircle, with centre O. P is 2 cm from O on the figure’s axis of symmetry. Points X and Y are fixed but the position of Z is variable on the shape’s perimeter.a. Find the mean distance from P to X Y, and Z when angle
The percentage scores of 40 candidates who took a Health and Safety test are given:77 44 65 84 52 60 35 83 68 66 50 68 65 57 60 50 93 38 46 5545 69 61 64 40 66 91 59 61 74 70 75 42 65 85 63 73 84 68 30a. Construct a frequency table by grouping the data into seven classes with equal-width intervals,
The frequency densities of the four classes in a histogram are in the ratio 4 : 3 : 2 :1. The frequencies of these classes are in the ratio 10 :15 : 24 : 8.Find the total width of the histogram, given that the narrowest class interval is represented by a column of width 3cm.
For 10 values denoted x, it is given that x̅ = 7.4. Find:a. Σxb. Σ(x + 2)c. Σ(x – 2)
The masses, xkg, of 12 objects are such that x̅ = 0.475. Find the value of 1000x and state what it represents.
The number of patients treated each day by a dentist during a 20-day period is shown in the following stem-and-leaf diagram.a. Find the median number of patients.b. On eight of these 20 days, the dentist arrived late to collect their son from school. If they decide to use their average number of
For each of the following sets of data, decide whether you would expect the mean to be less than, equal to or greater than the median and the mode.a. The ages of patients receiving long-term care at a hospital.b. The numbers of goals scored in football matches.c. The heights of adults living in a
Which of the eleven words in this sentence is the mode?
a. The mean of 15, 31, 47, 83, 97, 119 and p2 is 63. Find the possible values of p.b. The mean of 6, 29, 3, 14, q, (q + 8), q2 and (10 – q) is 20. Find the possible values of q.
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