A Level Mathematics Questionbanks



1. A bag contains three discs marked 1,2 and 3. A disc is drawn out and replaced and then another is drawn.

The two discs drawn are used to form a two digit number – for example, if one is drawn first and three is

drawn second, the number formed is 13.

a) Find the probability that the number formed is

i) prime

[2]

ii) greater than 13

[1]

b) Given that the two digit number is greater than 13, find the probability it is prime.

[4]

2. Twelve business people attend a lunch. The main course is served with a choice of carrots or broccoli or both.

Six people request just carrots and four people request broccoli.

a) State how many people had neither vegetable.

[1]

Given that the events “having broccoli” and “having carrots” are independent,

b) Find the number of people who had both vegetables.

[6]

3. a) Explain the meaning in words of the probability statement

P(A(B) = P(A) + P(B) – P(A(B)

[2]

P(A) = [pic]; P(B) =[pic]; P(A((B) = [pic]

b) Find P(A|B)

[3]

c) State the relationship between A and B

[1]

d) Calculate P(A((B(), and explain what is meant by this probability.

[3]

4. In the population, 0.2% have a particular disease. There is a test for this disease; all people who have the

disease give a positive test result, but 10% of those who do not have the disease also give a positive result.

a) Find the probability that a randomly selected member of the population tests positive for the disease.

[3]

b) Andy has just received a positive test result. Find the probability that he actually has the disease.

[4]

c) Comment on your answer to b)

[1]

5. A shop stocks three flavours of tinned cat food – Tuna, Chicken and Lamb – in the ratio 2:3:5.

Each flavour comes in two sizes, large and small. Twenty percent of Tuna tins are large; the corresponding

figures for Chicken and Lamb are 30% and 50%.

a) Find the probability that a randomly selected tin of cat food is large

[3]

b) A small tin is selected at random. Find the probability it is Lamb flavour.

[5]

c) Three tins are selected at random. Find the probability that they include exactly one of each flavour.

[3]

6. In a school, all girls play at least one of hockey and netball. If a girl plays netball, the probability that

she also plays hockey is 0.4. If a girl plays hockey, the probability that she also plays netball is 0.3. Find:

a) the probability that a randomly selected girl plays both sports

[9]

b) the probability that a girl selected at random plays hockey only

[2]

7. A bag contains 3 red marbles, 2 yellow marbles and 3 green marbles. Three marbles are selected at random

from the bag, without replacement. Find the probability that:

a) All are of the same colour

[4]

b) All are of different colours

[3]

c) Given that there are at least 2 red marbles, find the probability there are 3 red marbles

[6]

8. Anna and Deepa play three games of tennis. They are equally likely to win the first game.

If Anna wins one game, she has a probability of 0.6 of winning the next game. If Deepa wins one game,

she has a probability of 0.7 of winning the next game. Find the probability that:

a) Anna wins the third game

[4]

b) Deepa wins at least two games

[5]

c) Deepa wins at least two games, given that she wins at least one game

[5]

9. In a survey on married couples, it was found that 55% of the husbands could speak at least one foreign language.

If the husband could speak at least one foreign language, the probability that the wife could was [pic].

If the husband could not speak at least one foreign language, the probability that the wife could not was [pic].

A married couple was selected at random. Find the probability that

a) both could speak at least one foreign language

[3]

b) neither could speak any foreign languages

[3]

c) just the wife could speak at least one foreign language

[2]

Two married couples were selected at random.

d) Find the probability that just one of the husbands and just one of the wives were able to speak at

least one foreign language.

[5]

10. For two events, A and B, it is given that P(B) = [pic], P(A(|B) =[pic], P(A|B()= [pic]. Find the probability that:

a) B occurs and A does not

[2]

b) Just one of A and B occur

[5]

c) A occurs, given that B occurs

[3]

11. A bag contains 6 toffees, 5 caramels and 3 chocolates. A girl selects two sweets without replacement

Find the probability that

a) Both sweets were toffees

[2]

b) Both sweets were toffees, given that both sweets were the same

[6]

The girl then selects two more sweets, one at a time.

c) Find the probability that the fourth sweet was a toffee, given that the first two sweets were both toffees.

[4]

12. For events A and B, it is given that P(A) = 0.4 and P(B) = 0.3. Find P(A(B) if

a) A and B are mutually exclusive

[2]

b) A and B are independent

[4]

13. Two standard dice are thrown, one red and one blue. Event A occurs if the red dice shows a prime number.

Event B occurs if the total obtained by adding the numbers on each dice is between 7 and 10 inclusive.

a) Find the probabilities of events A and B

[4]

b) Find P(B|A)

[4]

c) State whether or not events A and B are independent, explaining your reasoning.

[2]

14. In a school staff room, the magazines “Private Eye” and “New Statesman” are available for reading.

2% of the staff read neither magazine, and 20% read both.

a) Write down the percentage of the staff who read just one magazine

[1]

Half of the staff who read “Private Eye” also read “New Statesman”

b) Find the percentage of the staff who read “Private Eye”

[2]

c) Given that a member of staff reads just one magazine, find the probability that it is “Private Eye”

[4]

15. Events A and B are independent, and satisfy P(A(B) = 0.58; P(A(B) =0.12. Let P(A)=x and P(B)=y.

a) Write down two equations satisfied by x and y

[3]

b) Given that A is more likely to occur than B, find P(A) and P(B)

[5]

c) State the value of P(A|B)

[1]

16. Bag A contains 3 red balls and 1 white ball. Bag B contains 2 red balls and 3 white balls.

Sofia takes a ball at random out of bag A, notes its colour, and places it into bag B.

She then draws a ball at random from bag B, notes its colour and places it into bag A.

R1 denotes the event that the first ball is red; R2, W1 and W2 are defined similarly.

a) Find P(W1(W2)

[3]

b) Find P(W2)

[4]

c) Find P(W1|R2)

[5]

17. A student sits modules P1 and M1 in an examination. The probability that he passes P1 is p.

If he passes P1, he has a 50% chance of passing M1. If he fails P1, he has a 30% chance of passing M1.

a) Find, in terms of p, the probability that the student passes M1

[3]

b) Show that the conditional probability that the student fails P1 given that he passes M1 is [pic]

[4]

c) Given that the probability calculated in b) is equal to [pic], find the value of p.

[3]

d) Find the probability that the student has passed P1, given that he has passed at least one examination

[6]

18. Anil, Emma and Toni share a student house. The probabilities that each of them is home on a given evening

are 0.3, 0.5 and 0.1 respectively. If a telephone call arrives in the evening, the probabilities that it is for each

of the students are 0.1, 0.7 and 0.2 respectively, independent of who is at home.

If a telephone call arrives on a randomly selected evening, find the probability that

a) the call is for Anil and he is at home.

[2]

b) the call is for someone who is at home

[3]

c) Given that the call is for someone who is at home, find the probability the call is for Toni.

[4]

There are telephone calls on three successive evenings

d) Find the probability there is one call for each student

[3]

19. Events Q and R are such that P(Q) = q and P(R(Q() = x

a) Explain what is meant by P(R(Q() and P(R((Q()

[2]

b) Find P(R((Q(), giving your answer in terms of q and x

[3]

c) Given that Q and R are independent, and that P(R(Q)=0.1, show that [pic]

[4]

d) Given, further, that x=0.3, find P(R((Q)

[3]

20. A business uses three different firms of couriers, A, B and C, with probabilities 0.4, 0.5 and 0.1 respectively.

Firm A delivers promptly on 75% of occasions; the corresponding figures for B and C are 80% and 70% respectively. A courier delivery is selected at random. Find the probability:

a) It is prompt

[3]

b) It is by firm B, given that it is not prompt

[5]

The business uses couriers four times in one week. Find the probability that:

c) Firm C is used at least once

[3]

d) Firm C is used exactly three times, given that it is used at least once.

[4]

21. It is known that it rains on 30% of days during August in a particular town.

In a week in August, find the probability that

a) It rains on Monday only during the (seven-day) week

[3]

b) There is only one day of the week on which it does not rain

[3]

c) State an assumption that has been used in the calculations in this question

[1]

22. A bag contains R red balls. There are 10 balls in total, and the remainder of the balls are green.

A second bag contains 2 red balls and 2 green balls. One ball is taken out of the first bag and placed in

the second bag. A ball is then selected at random from the second bag, and its colour is noted.

The probability that the ball selected from the second bag is red is [pic].

a) Show that R=4

[5]

b) Given that the ball selected from the second bag is red, find the probability that the ball selected

from the first bag was also red.

[3]

23. At a college, thirty percent of students study neither Art nor Biology. Thirty percent of students study Biology.

Twenty percent of Art students study Biology. Find the percentage of students:

a) who study both Art and Biology

[7]

b) who study exactly one of these two subjects

[3]

A particular student is known to study Biology.

c) Find the probability that she does not study Art

[3]

The college sets examinations in Biology and Art on a particular day.

d) Find the probability that if two students are selected at random, they have done precisely two

examinations between them on this day.

[5]

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