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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

Find the probability that each of the following events occur.a. Exactly five heads are obtained when a fair coin is tossed nine times.b. Exactly two 6s are obtained with 11 rolls of a fair die.
Weiqi has two fair triangular spinners. The sides of one spinner are labelled 1, 2, 3, and the sides of the other are labelled 2, 3, 4. Weiqi spins them simultaneously and notes the two numbers on
Let T be the number of times that a fair coin is tossed, up to and including the toss on which the first tail is obtained. Find the mode and the mean of T.
On average, Diya concedes one penalty in every six hockey matches that she plays. Find the probability that Diya next concedes a penalty:a. In the eighth match that she plays b. After the fourth
Given that X ~ B(n, p), E(X ) = 20 and Var(X ) = 12, find:a. The value of n and of pb. P(X = 21).
Given that B~B(8, 2/7), find:a. P(V = 4)b. P(V ≥ 7)c. P(V ≤ 2)d. P(3 ≤ V < 6)e. P(V is an odd number).
One plastic robot is given away free inside each packet of a certain brand of biscuits. There are four colours of plastic robot (red, yellow, blue and green) and each colour is equally likely to
Given that S ~ Geo( p) and that E(S) 4 1/2 = , find P(S = 2).
The probability that Mike is shown a yellow card in any football match that he plays is 1/2. Find the probability that Mike is next shown a yellow card:a. In the third match that he playsb. Before
Given that Y ~ B(11, 0.23), calculate:a. P(Y ≠ 3)b. P[Y < E(Y )].
Given that W ~B(9, 0.32), find:a. P(W = 5)b. P(W ≠ 5)c. P(W < 2)d. P(0
A family has booked a long holiday in Skragness, where the probability of rain on any particular day is 0.3.Find the probability that:a. The first day of rain is on the third day of their holidayb.
The random variable Y follows a geometric distribution. Given that P(Y = 1) = 0.2, find E(Y ).
Given that T ~ Geo(0.32), find:a. P(T = 3)b. P(T ≤ 6)c. P(T > 7) .
Given that X ~ B(8,0.25), calculate:a. E(X ) and Var(X )b. P[X = E(X )] c. P[X < E(X )].
Given that Y ~B(7, 0.6), find:a. P(Y = 7)b. P(Y = 5)c. P(Y ≠ 4)d. P(3
Given that X~ B(n, 1/n) find an expression for P(X = 1) in terms of n.
Given that X ~ Geo(0.36), find the exact value of E(X ).
Given the discrete random variable X ~ Geo(0.2), find:a. P(X = 7)b. P(X ≠ 5)c. P(X > 4) .
Calculate the expectation, variance and standard deviation of each of the following discrete random variables. Give non-exact answers correct to 3 significant figures.a. V ~ B(5, 0.2)b. W ~ B(24,
The variable X has a binomial distribution with n = 4 and p = 0.2. Find:a. P(X = 4)b. P(X = 0)c. P(X = 3)d. P(X = 3 or 4).
The probability that a boy succeeds with each basketball shot is 7/9. He takes two shots and the discrete random variable S represents the number of successful shots.Show that P(S = 0) = 4/81 and
Five grapes are randomly selected without replacement from a bag containing one red grape and six green grapes.Name and list the possible values of two discrete random variables in this
Find the mean and the variance of the discrete random variable X, whose probability distribution is given in the following table. 1 2 3 P(X =x) 1-k 2- 3k 3-4k 4- 6k %3D
The discrete random variable V is such that V ∈{1, 2, 3}. Given that P(V = 1) = P(V = 2) = 2 × P(V = 3), draw up the probability distribution table for V.
The probability distribution for the random variable X is given in the following table.Calculate E(X ) and Var(X ). 2 3 P(X = x) 0.10 0.12 0.36 0.42
The following table shows the probability distribution for the random variable Y.a. Given that Var(Y) = 1385.2, show that q2 – 61q + 624 = 0 and solve this equation.b. Find the greatest possible
The probability distribution for the random variable X is given in the following table.Find the value of p and work out P(2 < X < 5). 2. 3 4 P(X = x) P 2p 3p %3D in
The probability distribution for the random variable Y is given in the following table.a. Find the value of p.b. Calculate E(Y ) and the standard deviation of Y. 1 2 3 4 P(Y= 0.03 2p 0.32 0.05
An investment company has produced the following table, which shows the probabilities of various percentage profits on money invested over a period of 3 years.a. Calculate the expected profit on an
The probability distribution for the random variable W is given in the following table.a. Form an equation using k, then solve it.b. Explain why only one of your solutions is valid.c. Find P(6 ≤
The random variable T is such that T ∈{1, 3, 6,10}. Given that the four possible values of T are equiprobable, find E(T) and Var(T).
A chef wishes to decorate each of four cupcakes with one randomly selected sweet. They choose the sweets at random from eight toffees, three chocolates and one jelly. Find the variance of the number
The following table shows the probability distribution for the random variable V.Given that E(V) = 5.38, find the value of m and calculate Var(V). 1 3 9. P(V = v) 0.4 0.28 0.14 0.18
The faces of a biased die are numbered 1, 2, 3, 4, 5 and 6. The random variable X is the score when the die is thrown. The probability distribution table for X is given.The die is thrown 3 times.
R is a random variable such that R ∈{10, 20, 70,100}. Given that P(R = r) is proportional to r, show that E(R) = 77 and find Var(R).
At a garden centre, there is a display of roses: 25 are red, 20 are white, 15 are pink and 5 are orange. Three roses are chosen at random.a. Show that the probability of selecting three red roses is
A picnic basket contains five jars: one of marmalade, two of peanut butter and two of jam. A boy removes one jar at random from the basket and then his sister takes two jars, both selected at
Three vehicles from a company’s six trucks, five vans, three cars and one motorbike are randomly selected and tested for roadworthiness.a. Show that the probability of selecting three vans is
The probability distribution for the random variable W is given in the following table.Given that E(W) = a, find a and evaluate Var(W). 7 a 24 P(W = 0.3 0.3 ab 0.1 0.3
Two ordinary fair dice are rolled. The product and the sum of the two numbers obtained are calculated.The score awarded, S, is equal to the absolute (i.e. non-negative) difference between the product
The possible outcomes from a business venture are graded from 5 to 1, as shown in the following table.a. Calculate the expected grade and use it to describe the expected outcome of the venture. Find
A fair triangular spinner has sides labelled 0,1 and 2, and another fair triangular spinner has sides labelled –1, 0 and 1. The score, X, is equal to the sum of the squares of the two numbers on
A pack of five DVDs contains three movies and two documentaries. Three DVDs are selected and the following table shows the probability distribution for M, the number of movies selected.Draw up the
Two ordinary fair dice are rolled. The discrete random variable X is the lowest common multiple of the two numbers rolled.a. Draw up the probability distribution table for X.b. Find E(X ) and P[X
A discrete random variable X, where X ∈{2, 3, 4, 5}, is such that P(X = x) = (b – x)2/30a. Calculate the two possible values of b.b. Hence, find P(2 < X < 5).
In a particular country, 90% of the population is right-handed and 40% of the population has red hair. Two people are randomly selected from the population. Draw up the probability distribution for
In a game, a player attempts to hit a target by throwing three darts. With each throw, a player has a 30% chance of hitting the target.a. Draw up the probability distribution table for H, the number
Two students are randomly selected from a class of 12 girls and 18 boys.a. Find the expected number of girls and the expected number of boys.b. Write the ratio of the expected number of girls to the
Set A consists of the ten digits 0, 0, 0, 0, 0, 0, 2, 2, 2, 4.Set B consists of the seven digits 0, 0, 0, 0, 2, 2, 2.One digit is chosen at random from each set. The random variable X is defined as
A fair 4-sided die, numbered 1, 2, 3 and 5, is rolled twice. The random variable X is the sum of the two numbers on which the die comes to rest.a. Show that P(X = 8) = 1/8.b. Draw up the probability
A sewing basket contains eight reels of cotton: four are green, three are red and one is yellow. Three reels of cotton are randomly selected from the basket.a. Show that the expected number of yellow
The discrete random variable Y is such that Y ∈{4, 5, 8,14,17} and P(Y = y) is directly proportional to 1/y + 1. Find P(Y > 4).
There are eight letters in a post box, and five of them are addressed to Mr Nut. Mr Nut removes four letters at random from the box.a. Find the probability that none of the selected letters are
A company offers a $1000 cash loan to anyone earning a monthly salary of at least $2000. To secure the loan, the borrower signs a contract with a promise to repay the $1000 plus a fixed fee before 3
X is a discrete random variable and X ∈{0, 1, 2, 3}. Given that P(X > 1) = 0.24, P(0 < X < 3) = 0.5 and P(X = 0 or 2) = 0.62, find P(X ≤ 2 |X > 0).
A discrete random variable Y is such that Y ∈{8, 9, 10}. Given that P(Y = y) = ky, find the value of the constant k.
When a scout group of 8 juniors and 12 seniors meets on a Monday evening, one scout is randomly selected to hoist a flag. Let the variable X represent the number of juniors selected over n
Four students are to be selected at random from a group that consists of seven boys and x girls. The variables B and G are, respectively, the number of boys selected and the number of girls
Q is a discrete random variable and Q ∈{3, 4, 5, 6}.a. Given that P(Q = q) = cq2, find the value of the constant c.b. Hence, find P(Q > 4).
An ordinary fair die is rolled. If the die shows an odd number then S, the score awarded, is equal to that number. If the die shows an even number, then the die is rolled again. If on the second roll
A box contains 2 green apples and 2 red apples. Apples are taken from the box, one at a time, without replacement. When both red apples have been taken, the process stops. The random variable X is
Four books are randomly selected from a box containing 10 novels, 10 reference books and 5 dictionaries.The random variable N represents the number of novels selected.a. Find the value of P(N = 2),
A fair 4-sided spinner with sides labelled A, B, B, B is spun four times.a. Show that there are six equally likely ways to obtain exactly two Bs with the four spins.b. By drawing up the probability
In a particular discrete probability distribution the random variable X takes the value 120/r with probability r/45, where r takes all integer values from 1 to 9 inclusive.i. Show that P(X = 40) =
In a game, a fair 4-sided spinner with edges labelled 0, 1, 2 and 3 is spun. If a player spins 1, 2 or 3, then that is their score. If a player scores 0, then they spin a fair triangular spinner with
A biased coin is tossed three times. The probability distribution for H, the number of heads obtained, is shown in the following table.a. Find the probability of obtaining a head each time the coin
Two ordinary fair dice are rolled. A score of 3 points is awarded if exactly one die shows an odd number and there is also a difference of 1 between the two numbers obtained. A player who rolls two
The discrete random variable R is such that R ∈{1, 3, 5, 7}a. Given that P(R = r) = k(r + 1)/r + 2, find the value of the constant k.b. Hence, find P(R ≤ 4).
Find how many of the arrangements of four letters from A, B, C, D, E and F:a. Begin with the letter A b. Contain the letter A.
A book of poetry contains seven poems, three of which are illustrated. In how many different orders can all the poems be read if no two illustrated poems are read one after the other?
Find the number of ways that seven goats and four sheep can sleep in a row if:a. All the goats must sleep next to each otherb. No two sheep may sleep next to each other.
A radio presenter has enough time at the end of their show to play five songs. She has 13 songs by four groups to choose from: five songs by The Anvils, four by The Braziers, three by The Chisels and
Students enrolling at an A Level college must select three different subjects to study from the six that are available. One subject must be chosen from each of the option groups A, B and C, as shown
A teacher is looking for 6 pupils to appear in the school play and has decided to select them at random from a group of 11 girls and 13 boys.a. Find the number of ways in which the teacher can select
Five people are randomly selected from a group of 67 women and 33 men. Find the probability that the selection consists of an odd number of women.
Four ordinary fair dice are arranged in a row. Find the number of ways in which this can be done if the four numbers showing on top of the dice:a. Are all oddb. Have a sum that is less than 7.
At company V, 12.5% of the employees have a university degree. At company W, 85% of the employees do not have a university degree. There are 112 employees at company V and 120 employees at company
One hundred qualified drivers are selected at random. Out of these 100 drivers, of the 40 drivers who wear spectacles, 30 passed their driving test at the first attempt. Altogether, 25 of the drivers
A conference hall has 24 overhead lights. Pairs of lights are operated by switches next to the main entrance, and each switch has three numbered settings: 0 (off), 1 (dim), 2 (bright). Find the
Twelve chairs in two colours are arranged, as shown.Find in how many ways nine people can sit on these chairs if:a. The two blue chairs in column C must remain unoccupiedb. All of the green chairs
In a certain country, vehicle registration plates consist of seven characters: a letter, followed by a three digit number, followed by three letters.For example: B 474 PQRThe first letter cannot be a
Seats for the guests at an awards ceremony are arranged in two rows of eight and ten, divided by an aisle, as shownSeats are randomly allocated to 18 guests.a. Find the probability that two
Find the number of distinct five-letter arrangements that can be made from:a. Two As and three Bs b. Two identical vowels and three Bsc. Two identical vowels and any three identical consonants.
Find the number of distinct arrangements that can be made from all the letters in the word THEATRE when the arrangement:a. Begins with two Ts and ends with two Es b. Has H as its middle letterc.
From a set of 10 rings, a jeweller wishes to display seven of them in their shop window. The formation of the display is shown in the diagram opposite. Find the number of possible displays if, from
A bag contains six red fuses, five blue fuses and four yellow fuses. Find how many ways there are to select:a. Three fuses of different colours b. Three fuses of the same colourc. 10 fuses in
Five clowns each have a red wig and a blue wig, which they are all equally likely to wear at any particular time. Find the probability that, at any particular time:a. Exactly two clowns are wearing
Find how many ways 15 children can be divided into three groups of five if:a. There are no restrictionsb. Two of the children are brothers who must be in the same group.
The following diagram shows a row of post boxes with the owners’ names beneath.Five parcels, one for the owner of each box, have arrived at the post office. If one parcel is randomly placed in each
Using each digit not more than once, how many even four-digit numbers can be made from the digits 1, 2, 3, 4, 5, 6 and 7?
The diagram opposite shows the activities offered to children at a school camp.If children must choose three activities to fill their day, how many sets of three activities are there to choose from?
A gardener has nine rose bushes to plant: three have red flowers and six have yellow flowers. If they plant them in a row in random order, find the probability that:a. A yellow rose bush is in the
An entertainer has been asked to give a performance consisting of four items. They know three songs, five jokes, two juggling tricks and can play one tune on the mandolin. Find how many different
There are x boys and y girls to be arranged in a line. Find the relationship between x and y if it is not possible to separate all the boys.
Find how many three-digit numbers can be made from the digits 0, 1, 2, 3 and 4, used at most once each, if the three-digit number:a. Must be a multiple of 10 b. Cannot begin with zero.
A fair triangular spinner with sides numbered 1, 2 and 3 is spun three times and the numbers that it comes to rest on are written down from left to right to form a three-digit number.a. How many
Two taxis are hired to take a group of eight friends to the airport. One taxi can carry five passengers and the other can carry three passengers.What information is given in this situation by the
A farmer has 50 animals. They have 24 sheep, of which three are male, and they have 26 cattle, of which 20 are female. A veterinary surgeon wishes to test six randomly selected animals. Find the
From a group of nine people, five are to be chosen at random to serve on a committee. In how many ways can this be done if two particular people refuse to serve on the committee together?

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