1. A group of 25 females were asked how many children they each had. The results are shown in
the histogram below.
Number of Children per Female
0 1 2 3 4
Number of Children
(d) A female is selected at random. What is the probability that she has more than two
(e) Two females are selected at random. What is the probability that:
(i) both females have more than two children?
(ii) only one of the females has more than two children?
(iii) the second female selected has two children given that the first female selected had
2. Nene and Deka both play netball. The probability that Nene will score a goal on her first attempt
is 0.75. The probability that Deka will score a goal on her first attempt is 0.82.
Calculate the probability that
(a) Nene and Deka will both score a goal on their first attempts;
(b) neither Nene nor Deka will score a goal on their first attempts.
3. In a club with 60 members, everyone attends either on Tuesday for Drama (D) or on Thursday
for Sports (S) or on both days for Drama and Sports.
One week it is found that 48 members attend for Drama and 44 members attend for Sports and x
members attend for both Drama and Sports.
(a) (i) Draw and label fully a Venn diagram to illustrate this information.
(ii) Find the number of members who attend for both Drama and Sports.
(iii) Describe, in words, the set represented by (D S)'.
(iv) What is the probability that a member selected at random attends for Drama only or
The club has 28 female members, 8 of whom attend for both Drama and Sports.
(b) What is the probability that a member of the club selected at random
(i) is female and attends for Drama only or Sports only?
(ii) is male and attends for both Drama and Sports?
4. Members of a certain club are required to register for one of three games, billiards, snooker,
darts, or fencing.
The number of club members of each gender choosing each game in a particular year is shown
in the table below.
Billiards Snooker Darts Fencing
Male 4 15 8 10
Female 10 21 17 37
A club member is to be selected at random.
(a) What is the probability that the club member selected is a
(i) female who chose billiards or snooker?
(ii) male or female who chose darts or fencing?
5. It is known that 5% of all AA batteries made by Power Manufacturers are defective. AA
batteries are sold in packs of 4.
Find the probability that a pack of 4 has
(a) exactly two defective batteries;
(b) at least one defective battery.
6. Events A and B have probabilities P(A) = 0.4, P (B) = 0.65, and P(A B) = 0.85.
(a) Calculate P(A B).
(b) State with a reason whether events A and B are independent.
(c) State with a reason whether events A and B are mutually exclusive.
7. The Venn diagram below shows the number of students studying Science (S), Mathematics (M)
and History (H) out of a group of 20 college students. Some of the students do not study any of
these subjects, 8 study Science, 10 study Mathematics and 9 study History.
S 1 M
(a) (i) How many students belong to the region labelled A?
(ii) Describe in words the region labelled A.
(iii) How many students do not study any of the three subjects?
(b) Draw a sketch of the Venn diagram above and shade the region which represents S H.
(c) Calculate n(S H).
This group of students is to compete in an annual quiz evening which tests knowledge of
Mathematics, Science and History. The names of the twenty students are written on pieces of
paper and then put into a bag.
(d) One name is randomly selected from the bag. Calculate the probability that the student
(i) all three subjects;
(ii) History or Science.
(e) A team of two students is to be randomly selected to compete in the quiz evening. The
first student selected will be the captain of the team. Calculate the probability that
(i) the captain studies all three subjects and the other team member does not study any
of the three subjects;
(ii) one student studies Science only and the other student studies History only;
(iii) the second student selected studies History, given that the captain studies History
8. Today Philip intends to go walking. The probability of good weather (G) is . If the weather is
good, the probability he will go walking (W) is . If the weather forecast is not good (NG)
the probability he will go walking is .
(a) Complete the probability tree diagram to illustrate this information.
(b) What is the probability that Philip will go walking?
9. Amos travels to school either by car or by bicycle. The probability of being late for school is
if he travels by car and if he travels by bicycle. On any particular day he is equally
likely to travel by car or by bicycle.
(a) Draw a probability tree diagram to illustrate this information.
(b) Find the probability that
(i) Amos will travel by car and be late.
(ii) Amos will be late for school.
(c) Given that Amos is late for school, find the probability that he travelled by bicycle.
Hint: You have to use the P(A|B) formula!
10. The table below shows the relative frequencies of the ages of the students at Ingham High
(in years) frequency
(a) If a student is randomly selected from this school, find the probability that
(i) the student is 15 years old;
(ii) the student is 16 years of age or older.
There are 1200 students at Ingham High School.
(b) Calculate the number of 15 year old students.
11. Two jars contain a number of coloured balls as indicated in the diagrams below.
2 Black 2 Black
3 White 1 White
Jar One Jar Two
Two experiments are carried out.
First Experiment: A jar is first chosen at random and then a ball is drawn from that jar.
(a) Draw, and label fully, a tree diagram to show all possible outcomes of this experiment.
(b) What is the probability that a white ball is drawn?
Second Experiment: The ball drawn in the first experiment is not replaced. A second ball is then
drawn from the same jar.
(c) What is the probability that both balls are white?
(Total 7 marks)
12. On a particular day 100 children are asked to make a note of what they drank that day.
They are given three choices: water (W), coffee (C) or fruit juice (F)
1 child drank only water.
6 children drank only coffee.
8 children drank only fruit juice.
5 children drank all three.
7 children drank water and coffee only.
53 children drank coffee and fruit juice only.
18 children drank water and fruit juice only.
(a) Represent the above information on a Venn Diagram.
(b) How many children drank none of the above?
(c) A child is chosen at random. Find the probability that the child drank
(ii) water or fruit juice but not coffee;
(iii) no fruit juice, given that the child did drink water.
(d) Two children are chosen at random. Find the probability that both children drank all three
(Total 13 marks)
13. At a certain school there are 90 students studying for their IB diploma. They are required to
study at least one of the subjects: Physics, Biology or Chemistry.
50 students are studying Physics,
60 students are studying Biology,
55 students are studying Chemistry,
30 students are studying both Physics and Biology,
10 students are studying both Biology and Chemistry but not Physics,
20 students are studying all three subjects.
Let x represent the number of students who study both Physics and Chemistry but not Biology.
Then 25–x is the number who study Chemistry only.
The figure below shows some of this information and can be used for working.
U with n(U) 90
(a) Express the number of students who study Physics only, in terms of x.
(b) Find x.
(c) Determine the number of students studying at least two of the subjects.