Download Statistics and probability: Chance

Statistics and probability: Chance
Chapter 13: Probability
Test B
Name: ___________________________
FLUENCY
Mark
1
Describe these events as either:
impossible,
likely,
or not likely.
a
b
c
3
You roll a die and the number 8 comes up.
You will be tested for your driver's licence at some stage.
You will have no take-away food for a week.
Answers:
a Impossible
b Likely
c Not likely
2
A die is rolled 50 times with the number 3 coming up 15 times. Find the relative frequency
of:
a
obtaining a 3 (as a percentage)
b
not obtaining a 3 (as a percentage).
4
Answers:
15
 100%
a
50
= 30%
b
35
 100%
50
= 70%
© John Wiley & Sons Australia, Ltd
1
Chapter 13: Probability
3
The estimated probability for getting a Heads when a coin is tossed is:
A 0
B 1
1
C
4
1
D
3
1
E
2
Test B
1
Answer: E
4
A 4-sector spinner was spun 10 times and the results noted as shown below.
3, 4, 3, 1, 2, 3, 2, 1, 1, 1
a
Organise the data into a frequency table.
b
Find the relative frequency of each outcome.
c
What is the total of all the relative frequencies?
Answers:
a
Score
1
2
3
4
5
6
Frequency
4
2
3
1
4
10
2
RF(2) =
10
3
RF(3) =
10
1
RF(4) =
10
b
RF(1) =
c
1
The sample space when rolling a die is:
A {2, 4, 6}
B {0, 1, 2, 3, 4}
C {3, 6, 9}
D {4, 8, 12}
E {1, 2, 3, 4, 5, 6}
1
Answer: E
© John Wiley & Sons Australia, Ltd
2
Chapter 13: Probability
6
A standard packet of 52 playing cards has 4 suits of 13 cards each. The 4 suits are hearts,
clubs, diamonds and spades. The cards in each suit are labelled 2 to 10, and then jack,
queen, king and ace. A card is drawn at random (and is replaced each time).
What is the probability of getting:
a
a 3?
b
a queen?
c
an odd number (an ace is taken as one)?
d
a jack of diamonds?
Answers:
4
=
52
a
4
=
52
b
7
Test B
4
1
13
1
13
c
20 5
=
52 13
d
1
52
A standard die is rolled. Find the probability of getting:
a
a6
b
2 or 6
c
an even number
d
a number less than 4
e
a multiple of 3.
5
Answers:
1
6
a
b
2
1
=
6
3
c
1
2
d
1
2
e
2 1

6 3
© John Wiley & Sons Australia, Ltd
3
Chapter 13: Probability
8
The tree diagram below shows the event of tossing a coin then spinning a 4 sector spinner.
Test B
1
The outcomes are:
A H123456
B HHHH1234
C 1234
D H1 H2 H3 H4 T1 T2 T3 T4
E HT
Answer: D
9
Draw a Venn diagram to represent the relationship between the following sets. Show the
position of all the elements in the Venn diagram.
ξ = {p: p is a positive integer less than 12}
F = {factors of 12}
E = {even numbers}
M = {multiples of 10}
4
Answer:
10
Use the diagram in question 9 to list the elements in the following sets.
a
E′
M E
b
FE
c
(M  E)  F
d
e
If a positive integer less than 12 is selected at random, what is the probability that it is
not a factor of 12, an even number or a multiple of 10?
5
Answers:
a
{1, 3, 5, 7, 9, 11}
b
{10}
c
{1, 2, 3, 4, 6, 8, 10}
d
{2, 4, 6}
e
© John Wiley & Sons Australia, Ltd
4
Chapter 13: Probability
Test B
UNDERSTANDING
11
If you rolled a die 80 times and the relative frequency of obtaining the number 3 was 0.2,
how many times would the number 3 have come up?
2
Answer:
Frequency of the event
= relative frequency  total number of trials
= 80  0.2
= 16
12
The probability of randomly choosing a type of book is
2
3
for a cookbook,
for an art
5
10
6
1
for a cartoon book. If there is a selection of 30 books, how many were
5
originally:
a
cookbooks
b
art books
c
cartoon books ?
book and
Answers:
a
2
 30 = 12
5
b
3
 30= 9
10
c
1
 30= 6
5
© John Wiley & Sons Australia, Ltd
5
Chapter 13: Probability
13
Beth chooses a marble from a bag containing 1 red and 2 blue marbles. The colour is noted,
it is replaced and a 2nd marble is selected. Draw a tree diagram or a two-way table to find
the probability of drawing:
a
a red then a blue marble
b
two blue marbles
c
no blue marbles
d
at least one blue marble.
Test B
4
Answers:
2
9
a
4
9
b
1
9
c
8
9
d
© John Wiley & Sons Australia, Ltd
6
Chapter 13: Probability
14
Mary is pregnant with her first child. There is equal chance of the child being a boy or girl.
The child’s eye colour will be either blue, brown or hazel (each colour is equally probable).
What is the probability that her child will be:
a
a blue-eyed boy?
b
not be hazel-eyed?
c
have brown eyes?
Test B
4
Answers:
1
6
a
b
c
15
4 2
=
6 3
2 1
=
6 3
The spinner is spun and a coin is tossed.
a
b
2
3
1
4
2
Draw a two-way table to
show all the possible outcomes.
What is the probability of obtaining a heads and a number greater than 3?
Answers:
a
H
T
1
H1
T1
2
H2
T2
3
H3
T3
4
H4
T4
b
© John Wiley & Sons Australia, Ltd
7
Chapter 13: Probability
Test B
REASONING
16
Gabby is looking for someone to study French with. She wants to find a class member who
lives near her home and studies French. The form teacher tells her that the relative frequency
of French students in the class is 0.5 and the relative frequency of class members who live
near Gabby is 0.8. Gabby believes her teacher has made an error as these two relative
frequencies add to greater than 1. Is Gabby correct? Explain your answer.
2
Answer:
No, Gabby is incorrect. As it is possible for a student to both live near Gabby and study
French the total of these relative frequencies can be greater than 1.
17
Tess and Dane are playing a game with a 100-sided die. Dane gets one point if he rolls an
odd number greater than 50, while Tess gets three points if she rolls a number divisible by
10. Is the game a fair one? Explain your answer.
4
Answer:
For Dane,
25
100
1
Pr (odd > 50) =
4
For Tess,
P (odd > 50) =
20
100
1
Pr (divisible by 10) =
5
Pr(divisible by 10) =
In 100 rolls, we would expect Dane to get an odd number greater than 50 twenty-five times.
25 × 1 = 25 points in 100 rolls.
In 100 rolls, we would expect Tess to get a number divisible by 10 ten times.
10 × 3 = 30 points in 100 rolls.
The game is not fair, it favours Tess.
18
In a class of 29 students all students study either piano or guitar or both. If 20 study piano
and 12 study both, how many students study only guitar?
2
Answer:
Of the 20 that study piano some of these study guitar as well. As all students study at least
one of piano or guitar, the number of students studying only guitar is the total number of
students minus those that study piano only and study both.
29 – 20 = 9 study only guitar.
© John Wiley & Sons Australia, Ltd
8
Chapter 13: Probability
Test B
PROBLEM SOLVING
19
Mary is pregnant with her first child. There is equal chance of the child being either a boy
or girl. The child’s eye colour will be either blue, brown or hazel (each colour is equally
probable). What is the probability that the child will be female with blue or hazel eyes?
2
Answer:
2 1
=
6 3
20
Two people are to be selected from a group of 12 to participate in a quiz show. Adam and
Tamsin are amongst the 12 people. What is the probability that Adam and Tamsin will be
the two students selected.
3
Answer:
12  11 = 132 possible combinations
Adam could be selected first then Tamsin or Tamsin selected first then Adam, so two
desired combinations.
P(Tamsin and Adam selected)
=
=
© John Wiley & Sons Australia, Ltd
9