1
2
A card is drawn from a pack of 52 playing
cards. What is the probability the card drawn is:
26
52
1

2
(a) a red card?
(a) P(red) 
(b) a heart?
(b) P(heart) 
(c) an ace?
(c) P(ace) 
13
52
1

4
4
52
1

13
Two dice are rolled.
What is the probability that:
(a) the first die shows a 6?
(b) the second die shows an odd number?
(a) P(6) 
1
6
3
6
1

2
(b) P(odd) 
© John Wiley & Sons Australia, Ltd
Page 1
3
120 Year 10 students are surveyed on subject
satisfaction.
90 students like Maths and 105 students like
English.
80 students like both subjects.
What is the probability that a student selected
at random likes:
English only = 105 – 80
English only = 25
Maths only = 90 – 80
Maths only = 10
Neither Maths nor English = 120  80 – 25 – 10
Neither Maths nor English = 5
105
120
7
Pr (English =
8
(a) P(English) =
(a) English
25
120
5
Pr (English only=
24
(b) English only
(b) P(English only) =
(c) neither Maths nor English?
(c)
5
120
1
=
24
P(neither Maths nor English) =
(neither Maths nor English
© John Wiley & Sons Australia, Ltd
Page 2
4
Write the following odds as probabilities:
(a) 4–1
(a) 4–1
1
4 1
1

5

(b)
(b) 2–5
2–5
5
25
5

7

(c)
(c) 1–1
5
1–1
1
11
1

2

Write the following probabilities as odds:
1
(a)
4
(a)
1
1

4 1 3
= 3–1
(b)
6
3
10
( b)
3
3

10 7  3
= 7–3
State the event that is complementary to each
of the following:
(a) a coin landing Heads
(a) A coin landing Tails
(b) rolling less than 4 with a die
(b) Rolling more than 3
(c) selecting an ace from a standard deck.
(c) Not selecting an ace
© John Wiley & Sons Australia, Ltd
Page 3
7
The weather bureau forecasts the chance of rain
for the next day as 60%. What is the probability
of it not raining the next day?
P(rain)  60%
60
100
P(not raining)  1  P(rain)

60
100
3
 1
5
2

5
 1
8
A set of cards is numbered
{1, 2, 3, 4 … 24, 25}. A card is selected at
random. Find:
(a) P(not an even number)
(b) P(not a multiple of 5).
(a) P(not an even number)
= 1  P(even number)
12
=1
25
13
=
25
(b) P(not a multiple of 5)
= 1  P(multiple of 5)
5
=1
25
4
=
5
© John Wiley & Sons Australia, Ltd
Page 4
9
A bag contains 30 balls of which 12 are red,
7 are blue and the rest black. Find:
Number of black balls = 30 – 12 – 7
Number of black balls = 11
(a) P(not black)
(a) P(not black) = 1 – P(black)
11
Pr (no black = 1 
30
19
Pr (no black =
30
(b) P(neither red nor black).
(b) P(neither red nor black)
= 1  P(red or black)
23
=1
30
7
=
30
or
P(neither red nor black) = P(blue)
7
=
30
10
A raffle has 500 tickets. Kylie, Nicholas, Sergei P(not winning) = 1  P(winning)
and Damian purchased 120 tickets between
120
=1
them. What is the probability that none of them
500
wins?
19
=
25
© John Wiley & Sons Australia, Ltd
Page 5
1
A die is rolled. What is the probability that the
number is an odd number or a 4?
2
A set of cards is numbered {1, 2, 3 … 16}.
A card is selected at random. Find:
(a) P(multiple of 5 or a multiple of 6)
(b) P(a number less than 7 or greater than 8).
3 1

6 6
4

6
Pr(odd or a
2

3
P(odd  4) =
3 2
16
5
Pr(multiple of 5 or 6) 
16
(a) P(multiple of 5  6) =
(b) P(< 7 > 8) 
68
16
14
16
7

8

3
A card is selected at random from a pack of
52 playing cards. Find:
(a) P  a heart or a black card 
(a) P(heart  black) =
(b) P  an ace or a picture card (J,Q,K)  .
(b) P(ace  picture) =
13  26
52
39

52
Pr(heart or black)
3

4
4  43
52
16

52
Pr(ace or royal)
4

13
© John Wiley & Sons Australia, Ltd
Page 6
4
In a harness race, the odds of two horses
1
winning are, respectively, 5–1 and 2–1. What is 5–1  5  1
the probability that either horse wins?
1
5–1 =
6
1
2 1
1
2–1 =
3
2–1 
1 1

6 3
1 2
 
6 6
3
Pr(either horse wins) 
6
1

2
P(either horse wins) 
© John Wiley & Sons Australia, Ltd
Page 7
5
A card is selected at random from a pack of
52 playing cards. Find:
(a) P  a diamond or a red card 
(b) P  a queen or a black card  .
(a) P  diamond  red card 
 P(diamond)  P(red card)
 P(red card  diamond)
13 26 13



52 52 52
26

52
1

2
(b) P  queen  black card 
 P(queen)  P(black card)
 P(queen  black)
4 26 2



52 52 52
28

52
7

13
© John Wiley & Sons Australia, Ltd
Page 8
7
Two fair coins are tossed. Draw a tree diagram
and find the probability of tossing:
1
4
(a) 2 Heads
(a) P(2 Heads) =
(b) a Head and a Tail.
(b) P(Head  Tail) = P(HT) + P(TH)
1
1
Pr(Head and a T = +
4
4
1
Pr(Head and a ) =
2
A coin is tossed and a die is rolled.
(a) Draw a two-way table to show the sample
space.
(a)
Coin outcomes
6
Die outcomes
1
2
3
4
5
6
H
(H, 1) (H, 2) (H, 3) (H, 4) (H, 5) (H, 6)
T
(T, 1) (T, 2) (T, 3) (T, 4) (T, 5) (T, 6)
(b) Find the probability of getting a Tail with a (b) P(Tail > 4) = P(5, T) + P(6, T)
2
number greater than 4.
Pr(Tail and n ) =
12
1
Pr(Tail and n ) =
6
© John Wiley & Sons Australia, Ltd
Page 9
8
Two dice are rolled. Use a tree diagram to find
the probability of rolling:
(a) two sixes
(a) P(two sixes) = P(SS)
1
Pr(two sixes) =
36
(b) one six
(b) P(one six) = P(SS′) + P(S′S)
5
5
Pr(one six) =
+
36
36
5
Pr(one six) =
18
(c) at least one six.
(c) P(at least one six) = 1 – P(no sixes)
= 1  P(S′S′)
25
=1
36
11
=
36
© John Wiley & Sons Australia, Ltd
Page 10
9
A 2-digit number is to be formed using the
digits 1, 2 and 3. If the same number can be
used twice, find the probability that the number
formed is:
1
9
(a) 31
(a) P(31) =
(b) greater than 30
(b) P(>30) = P(31) + P(32) + P(33)
1
1
1
Pr(>30) = + +
9
9
9
1
Pr(>30) =
3
(c) odd.
(c) P(odd) = P(11) + P(13) + P(21)
+ P(23) + P(31) + P(33)
1
1
1
1
1
1
= + + + + +
9
9
9
9
9
9
2
=
3
© John Wiley & Sons Australia, Ltd
Page 11
10
Repeat question 9 if the same digit cannot be
used twice.
(a) P(31) =
1
6
(b) P(>30) = P(31) + P(32)
1
1
Pr(>30) = +
6
6
1
Pr(>30) =
3
(c) P(odd) = P(11) + P(21) + P(23) + P(31)
1
1
1
1
Pr(odd) = + + +
6
6
6
6
2
Pr(odd) =
3
© John Wiley & Sons Australia, Ltd
Page 12
11
A coin is tossed and a die is rolled. What is the
probability of an even number on the die and
Tails on the coin?
12
An 8-sided die and a 4-sided die are rolled.
Find:
(a) P(total of 7)
(b) P(total is a multiple of 3).
13
P(even  Tails) = P(even)  P(Tails)
3 1
= 
6 2
3

12
Pr(even and a T)
1

4
(a) P(total of 7)
= P(6,1) + P(5,2) + P(4,3) + P(3,4)
1 1 1 1 1 1 1 1
       
8 4 8 4 8 4 8 4
1
1
1
1




32 32 32 32
4

32
1

8
(b) P(total is a multiple of 3)
= P(total 3) + P(total 6) + P(total 9) +
P(total 12)
= P(1, 2) + P(2, 1) + P(1, 5) + P(2, 4)
+ P(3, 3) + P(4, 2) + P(1, 8) +
P(2, 7) + P(3, 6) + P(4, 5) + P(4, 8)
1 1
 11  
8 4
11

32
A netball player can shoot a goal 6 times out of
6 6
every 10 throws. What is the probability of her P(2 goals) = 10  10
shooting 2 goals from the next 2 throws?
9
Pr (2 goals) =
25
© John Wiley & Sons Australia, Ltd
Page 13
14
A bag contains 30 apples of which 4 are rotten.
Two apples are selected. What is the
probability of selecting:
4
3
(a) P(2 bad apples) =

(a) 2 bad apples?
30 29
2
Pr (2 bad apples) =
145
(b) 1 good and 1 bad apple?
15
(b) P 1 good apple and 1 bad apple 
= P(good and bad) + P(bad and good)
26 4
4 26
=

+

30 29
30 29
104
104
=
+
870
870
104
=
435
Annette is late for work once every 10 days.
Glenda her assistant is late for work once every
5 days. What is the probability that:
9 4
(a) P(both on time) =

(a) they are both on time?
10 5
18
Pr (both on time) =
25
(b) only one of them is on time?
(b) P(one on time)
= P(Annette on time and Glenda late)
+ P(Annette late and Glenda on time)
9 1 1 4
  
10 5 10 5
13

50

© John Wiley & Sons Australia, Ltd
Page 14
16
A gambling game requires the gambler to
correctly match 2 balls drawn from a barrel
containing 99 numbered balls. What is the
probability the gambler correctly matches:
(a) both balls?
(b) one of the balls?
17
18
2 1

99 98
1
Pr (both balls) =
4851
(a) P(both balls) =
(b) P(one of the balls)
= P(first ball and not the second)
+ P(second ball and not the first)
2 97 97 2




99 98 99 98
194
194


9702 9702
194

4851
Campbell has a 0.6 chance of passing a spelling
test and a 0.8 chance of passing a numeracy
test.
(a)
Find the probability that Campbell passes (a) P(S  N) = 0.6 × 0.8
both his tests.
= 0.48
(b)
Find the probability that Campbell passes (b) P(passes only one test)
only one test.
= P(S, not N) + P(not S, N)
= (0.6 × 0.2) + (0.4 × 0.8)
= 0.12 + 0.32
= 0.44
In her drawer, Mai has 6 blue socks and 4 red
socks. Mai randomly selects 2 socks. What is
the probability of Mai selecting 2 socks with
the same colour?
P(same colour)
= P(2 blue socks) + P(2 red socks)
6 5 4 3
   
10 9 10 9
30 12


90 90
7

15
© John Wiley & Sons Australia, Ltd
Page 15
19
Two cards are randomly selected from a pack
of cards. What is the probability both cards are:
(a) black?
13 12

52 51
1
=
17
(b) clubs?
(b) P(both clubs) =
4 3

52 51
1
Pr (both ace =
221
(c) aces?
(c) P(both aces) =
2 1

52 51
1
=
1326
(d) P(both black aces) =
(d) black aces?
20
26 25

52 51
25
=
102
(a) P(both black) =
In a class of 28 students, 2 students are selected
at random. Mark and Elise are two students in
this class. What is the probability that:
(a) one of them is selected?
(a) P(one is selected)
= P(Mark selected and Elise not)
+ P(Elise not and Mark selected)
+ P(Elise selected and Mark not)
+ P(Mark not and Elise selected)
1 26 1 26 1 26
1 26
=

+

+

+

28 27 28 27 28 27 28 27
26
=
189
(b) both are selected?
2
1

28 27
1
Pr(both are selected =
378
(b) P(both are selected) =
© John Wiley & Sons Australia, Ltd
Page 16