M_BANK\YR12-2U\PROBABILITY.CAT
Probability
1)!
2)!
3)!
4)!
2U84-6iii
The game of ‘odd man out’ is played with 3 people, each flipping a single unbiased coin. All
flip their coins simultaneously and if one face is different from the other two, the owner is
‘odd man out’ and loses. What is the probability that there is an odd man out on any given
turn?†
3
« »
4
2U84-6iv
Box A contains 4 defective and 16 non-defective light bulbs. Box B contains one defective
and one non-defective light bulb. A fair die is rolled once. If the outcome is a one or a two,
then a bulb is selected at random from Box A. Otherwise a selection is made from Box B.
What is the probability that the selected bulb will be defective?†
2
« »
5
2U85-2i
7
3
The probability of relief from a cold with antibiotic A is
whilst with antibiotic B is .
4
10
One cold sufferer takes A, the other B. What is the probability that:
a.
Both are cured?
b.
Neither is cured?†
21
3
« a)
b)
»
40
40
2U85-4i
After a cricket competition the following table appeared in a cricket magazine:
TEAM
Australia
Sri-Lanka
West Indies
5)!
WON TOSS
45%
60%
55%
BATTED FIRST
60%
80%
50%
GAMES WON
50%
20%
80%
Use the above results to answer the following questions, give your answers as fractions in
their lowest terms:
a.
What is the probability that Australia won the toss, batted first and won a particular
game?
b.
What is the probability that the West Indies won the toss, batted first and lost a
particular game?
c.
What is the probability of Sri Lanka not batting first and winning a particular
game?†
1
27
11
« a)
b)
c)
»
25
200
200
2U86-2iii
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6)!
7)!
8)!
In Australia roulette is played on a wheel with 37 equal slots. The slots are numbered
0, 1, 2, ..., 36 and are randomly distributed around the wheel. The slots are coloured
alternately red and black except the 0 slot which is usually green. What is the probability
that:
a.
zero shows in one spin?
b.
a red shows in one spin?
c.
the number 13 does not show in 5 spins?†
36 5
1
18
« a)
b)
c) (
) »
37
37
37
2U87-6i
A company installs two shut down systems to switch off dangerous machines. It has been
found that the first system works on 95% of the occasions when it is needed, while the second
system works on 90% of the occasions when it is required. Calculate the probability that:
a.
the machines are switched off when a dangerous situation arises;
b.
both systems fail.†
« a) 0·995 b) 0·005 »
2U88-7i
In an opinion poll, it was found that 70% of the people living in the city favoured the
proposed road closures in the city centre. Draw a tree diagram and use it to find the
probability that, if 3 people from the city electoral roll were chosen at random and
interviewed:
a.
all three will be in favour of the closures;
b.
the majority will be in favour;
c.
not more than 2 will be against the closures.†
« a) 0·343 b) 0·784 c) 0·973 »
2U89-3c
A family consists of 4 children; Alan; Barry; Colin and Diane. Two children are chosen at
random to go on a shopping errand. Draw a tree diagram to show the possible shopping
pairs. Hence find the probability of having a shopping pair:
i.
of a boy and a girl;
ii.
with Alan in it but not Barry.†
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A
B
C
D
A
B
C
A
B
D
D
A
B
C
« 
9)!
11)!
i)
1
2
ii)
1
»
3
2U90-1f
The table below gives the classification of 20 students in a class by hair colour and weight:
Weight
Overweight
Ideal Weight
Total
10)!
C
D
Hair Colour
Brunette
Blonde
10
3
5
2
15
5
Total
13
7
20
A student is selected from this class at random. Find the probability that the student is blonde
and not overweight.†
1
«
»
10
2U90-5c
From a group of 3 women and 2 men a committee of 3 is to be selected at random. Using a
tree diagram or otherwise, find the probability that the committee will consist of:
i.
3 women;
ii.
a majority of women.†
1
7
« i)
ii)
»
10
10
2U91-4b
The spinner shown below is used in a game.
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2
1
3
4
It is spun twice and the score recorded after each spin. Find the probability that:
i.
in each of the two spins the result is 4;
ii.
the sum of the two spins is 4.†
« i)
12)!
13)!
14)!
1
3
ii)
»
16
16
2U92-6c
PIN numbers are used for electronic banking. They consist of 4 digits with no restrictions on
the 4 digits. (e.g. 5222, 8383, 0126, etc.). James remembered that the first two digits of his
PIN number were 4 and 7 but he forgot the last two digits.
i.
What is the probability that he randomly guessed both of the next two digits
correctly?
ii.
James knew that his PIN number was even. What is the probability that he can
guess the correct PIN number?†
1
1
« i)
ii)
»
50
100
2U93-5b
Lisa had 3 similar keys in her pocket. To open her front door she tried the keys at random.
She stopped trying when she opened the door. She did not try the same key twice. Find the
probability:
i.
the door opened when she tried the first key;
ii.
she tried all three keys before the door opened.†
1
1
« i)
ii) »
3
3
2U94-5b
1
2
3
4
1
2
3
Group A
Group B
A card is chosen at random from the four cards in group A, then a second card is chosen at
random from the three cards in group B. What is the probability that:
i.
both of the cards chosen are numbered 2?
ii.
exactly one of the two cards chosen is numbered 2?
iii.
the number on the card from group A followed by the number on the card from
group B will form a two digit number greater than 40?†
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« i)
15)!
16)!
17)!
18)!
5
1
1
ii)
iii) »
4
12
12
2U94-10a
A school softball team has a probability of 0·8 of losing or drawing any match and a
probability of 0·2 of winning any match.
i.
Find the probability of the team winning at least one of three consecutive matches.
ii.
What is the least number of consecutive matches the team must play to be 90%
certain it will win at least one match?†
« i) 0·488 ii) 11 »
2U95-5a
Justin is a talented soccer goalkeeper. The probability that he can stop a penalty shot at goal
3
is . During a match the opposition had 2 penalty shots at goal. What is the probability
4
Justin stopped:
i.
both shots.
ii.
at least 1 shot.†
9
15
« i)
ii)
»
16
16
2U95-10b
Of the 4000 population of a small Scottish town, 2500 people are descendants of the
Campbell clan, 1900 are descendants of the Donald clan while 300 are not descendants of
either clan. A member of the town was selected at random. What is the probability he is a
descendant of both the Donald and Campbell clans?†
7
«
»
40
2U96-4c
Dino and Chris used the spinner shown above to play a game. Dino spun the spinner twice
and added the results of the two spins to get his score. Chris then took his turn. The player
with the highest score won the game.
i.
ii.
iii.
1
2
4
3
Use a tree diagram or a sample space to show all the possible scores Dino could
have achieved when he played the game.
What is the probability that Dino scored 6 in the game?
Dino’s score was 6. What is the probability that Chris won the game?†
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1
2
3
4
« i)
19)!
20)!
1
Score
2
2
3
3
4
4
5
1
3
2
4
3
5
4
6
1
4
2
5
3
6
4
7
1
5
2
6
3
7
4
8
ii)
3
3
iii)
»
16
16
2U96-9b
A horticulturist found the probability that a planted tomato seedling will eventually bear fruit
was 0·85. He planted n seedlings.
i.
What is the probability that no seedlings will bear fruit?
ii.
How many seedlings must be planted to be at least 99% certain that at least one
seedling will bear fruit.†
« i) (0·15)n ii) 3 »
2U97-4a
Two cards are chosen at random from the four cards shown below.
1
0
1
i.
ii.
2
Using a tree diagram, or otherwise, list all the possible outcomes.
What is the probability that the sum of the numbers on the cards chosen is zero?†
1
« i) (1, 0), (1, 1), (1, 2), (0, 1), (0, 1), (0, 2), (1, 1), (1, 0), (1, 2), (2, 1), (2, 0), (2, 1) ii) »
6
21)! 2U97-9b
In a touch football competition between the Red team and the Gold team, on average the Red
team has won 3 games out of every 4 games.
i.
Find the probability that the Red team wins the next two games.
ii.
In the next three games, what is the probability that the Red team wins more games
than the Gold team?†
9
27
« i)
ii)
»
32
16
22)! 2U98-1d
Two dice, with numbers 0 to 5 on their faces are thrown. What is the probability that they
both show 4 on their uppermost face?†
1
«
»
36
23)! 2U98-5b
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A box has 4 Geography books and 3 Mathematics books in it. Two books are selected at
random from the box.
i.
Draw a tree diagram to show all the possible outcomes.
ii.
Find the probability that:
.
the two books are Mathematics books;
.
at least one of the books is a Geography book.†
1
G
2
4
7
G
1
2
2
3
7
M
G
3
M
1
« i)
24)!
25)!
26)!
27)!
3
M
ii) )
6
1
) »
7
7
2U98-10a
A number of electrical components are wired together in an alarm so that it will operate if at
least one of the components works. The probability that each one of these components will
work is 0·6.
i.
If an alarm had three components wired together, find the probability that at least
one of the components will work.
ii.
Find the minimum number of components that must be wired together to be 99%
certain that the alarm will operate.†
« i) 0·936 ii) 6 »
2U99-6b
In an opinion poll conducted at a school, it was found that 70% of the students favoured an
Olympic parade through the city before the opening of the Olympic Games. Three students
were selected at random from the school and interviewed separately. By drawing a tree
diagram, or otherwise, find the probability that:
i.
all three students favoured the parade.
ii.
at least one of the students favoured the parade.
« i) 0·343 ii) 0·973 »
2U99-9a
A die numbered 1 to 6 is rolled twice. The sum S of the numbers which appear uppermost on
the die is calculated.
i.
Find the probability that S is greater than 9.
ii.
It is known that 5 appears on the die at least once in the two throws. Find the
probability that S is greater than 9.†
1
1
« i)
ii)
»
6
12
2U00-4b
A box contains five blue, three yellow and eight red beads. Two beads are selected at random
from the box without replacement. Find the probability that:
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M_BANK\YR12-2U\PROBABILITY.CAT
i.
ii.
both beads are blue.
at most one of the beads are blue.†
« i)
28)!
1
11
ii)
»
12
12
2U00-10b
Kellie and Lachlan play a game where they each take turns at throwing two ordinary dice.
The winner is the first person to throw a double. Kellie throws first.
5
i.
Show that the probability that Lachlan wins the game on his first throw is
.
36
ii.
Show that the probability Lachlan wins the game on his first or second throw is
5
53
 4.
given by
36 6
iii.
Calculate the probability that Lachlan wins the game.†
« i) ii) Proof iii)
29)!
2U01-4b
1
30)!
5
»
11
1
2
3
3
3
3
Two cards are chosen at random and without replacement from the seven cards above. What
is the probability that
i.
both cards show a 1
ii.
the sum of the two numbers on the cards chosen is greater than 4?†
1
10
« i)
ii)
»
21
21
2U01-10b
U
A
O
E
I
31)!
The spinner shown above is used in a game. Once spun, it is equally likely to stop at any one
of the letters A, E, I, O or U.
i.
If the spinner is spun twice, find the probability that is stops on the same letter twice.
ii.
How many times must the spinner be spun for it to be 99% certain that it will stop
on the letter E at least once?†
1
« i)
ii) 21 »
5
2U02-7a
A bag contains 5 blue balls, 4 red balls, 2 yellow balls and 1 green ball. Three balls are
selected at random without replacement from the bag. Calculate the probability that
i.
the three balls drawn are blue,
ii.
the three balls drawn are of the same colour,
iii.
exactly two of the balls drawn are blue.†
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« i)
1
7
7
ii)
iii)
»
22
22
110
32)! 2U03-5a
In a bag are 20 marbles. The bag consists of 7 red marbles, 9 gold marbles and 4 blue marbles.
One marble is drawn from the bag and not replaced, and then a second marble is drawn. With
the aid of a tree diagram, or otherwise, find the probability of choosing:
i.
two gold marbles
ii.
marbles of different colour.†
18
127
« i)
ii)
»
95
190
33)! 2U04-8a
A certain soccer team has a probability of 0∙6 of winning a match and a probability of 0∙3 of
drawing a match.
i.
If this soccer team plays two matches, draw a tree diagram to show all possible
outcomes.
ii.
Find the probability of this soccer team winning at least one match out of the two
matches.
iii.
Find the probability of this soccer team not winning either of the two matches.†
0∙6 W
0∙6
W
0∙3
D
0∙1
35)!
0∙6
0∙3
0∙1
0∙6
0∙3
0∙1
D
L
W
D
L
W
D
L
« i)
34)!
L
0∙3
0∙1
ii) 0∙84 iii) 0∙16 »
2U05-7a
Nicole and Mariana play against each other, in the third round of the Australian Open. In this
tournament, the first player to win 2 sets wins the match. The probability that Nicole wins
any set is 70%.
i.
Find the probability that the game will last two sets only.
ii.
Find the probability that Nicole wins the match.
iii.
Find the probability that Mariana wins the match.†
29
98
27
« i)
ii)
iii)
»
50
125
125
2U06-5d
During qualification for the 2006 World Cup, the Socceroos goalkeeper, Mark, defended
many penalty shots at goal. In fact, the probability that he can stop a penalty shot at goal
3
is . During a particular match, the opposing team had three penalty shots at goal.
5
Using a tree diagram, find the probability that:
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i.
ii.
the goalkeeper will stop all shots at goal.
the goalkeeper will stop at least 1 shot at goal.†
« i)
†©CSSA OF NSW 1984 - 2006
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367
27
117
ii)
»
125
125