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mathematics
cambridge international as & a level mathematics probability & statistics

Questions and Answers of Cambridge International AS & A Level Mathematics Probability & Statistics

A bag contains black and white counters, and the probability of selecting a black counter is 1/6.a. What is the smallest possible number of white counters in the bag?b. Without replacement, three
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
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
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
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
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
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
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
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
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,
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
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
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
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
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 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
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
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
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)
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
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
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
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
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
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.
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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,
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
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
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
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
Homes in a certain neighbourhood have recently sold for $220000, $242000, $236000 and $3500000. A potential buyer wants to know the average selling price in the neighbourhood. Which of the mean,
The numbers of items returned to the electrical department of a store on each of 100 consecutive days are given in the following table.a. Write down the median.b. Is the mode a good value to use as
The following table shows the hourly rates of pay, in dollars, of a company’s employees.a. Is the mean a good average to use here? Give a reason for your answer.b. Find the mean rate of pay for the
A dataset of 20 values is denoted by x where Σ(x −1) = 58. Another dataset of 30 values is denoted by y where Σ(y − 2) = 36. Find the mean of the 50 values of x and y.
One of the units used to measure pressure is pounds per square inch (psi). The mean pressure in the four tyres of a particular vehicle is denoted by x̅psi. Given that 1 pound is approximately equal
A study was carried out on 60 electronic items to find the currents, x amperes, that could be safely passed through them at a fixed voltage before they overheat. The results are given in the two
As part of a data collection exercise, members of a certain school year group were asked how long they spent on their Mathematics homework during one particular week. The times are given to the
A train makes a non-stop journey from one city to another and back again each day. Over a period of 30 days, the mean number of passengers per journey is exactly 61.5. Exact one-way ticket prices
Students investigated the prices in dollars ($) 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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,
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
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
Which of the eleven words in this sentence is the mode?

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