Download Probability - Cambridge University Press

17
ES
Ch apter
N
AL
PA
G
Statistics and Probability
Probability
FI
In this chapter, we introduce probability, which deals with how likely it is that
something will happen. This is an area of mathematics with many diverse
applications.
The study of probability began in seventeenth-century France, when the two
great French mathematicians Blaise Pascal (1623–1662) and Pierre de Fermat
(1601–1665) corresponded about problems from games of chance. Problems
such as these continued to influence the early development of the subject.
Nowadays probability is used in areas ranging from weather forecasting and
insurance, where it is used to calculate risk factors and premiums, to predicting
the risks and benefits of new medical treatments.
42 8
Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400
17A
An introduction to
probability
Most people would agree with the following statements:
• It is certain that the sun will rise tomorrow.
• If I toss a coin, getting a head and getting a tail are equally likely.
Everyday language
PA
G
ES
• There is no chance of finding a plant that speaks English.
N
AL
Using everyday language to discuss probabilities can cause problems, because people do not always
agree on the interpretation of words such as ‘likely’, ‘probable’ and ‘certain’. Consider these two
examples:
1 Two farmers are discussing the prospects of getting a good wheat crop this year. Farmer Bill says,
‘I don’t think it is likely to rain for the next two weeks. I’m not going to plant wheat yet.’ Farmer
Tony says, ‘I reckon you’re wrong. I’m certain we’ll have rain. It can’t go on the way it has. I’m
getting the tractor out tomorrow.’
FI
2 Alanna is captain of the Platypus Netball Team. They are going to play the Echidnas, whose
captain is Maria. Each captain says to her team before the match, ‘I think we’ll probably win. Just
follow the plans we’ve practised all week.’ Alanna and Maria cannot both be right! It would be
futile to try to assign a probability that the Echidnas (or the Platypuses) are going to win on the
basis of what the captains told their teams.
On the other hand, there are many situations in which it would be useful to be able to measure how
likely, or unlikely, it is that an event will occur. We can do this in mathematics by using the idea
of probability, which we define as a number between 0 and 1 that we assign to any event we are
interested in.
A probability of 1 represents an event that is ‘certain’ or ‘guaranteed to happen’.
A probability of 0 represents an event that we would describe as ‘impossible’ or one that ‘cannot
possibly occur’.
1
An event that has a probability is as likely to occur as not to occur.
2
C h apt e r 1 7 P r o babi l it y
Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400
429
1 7 A A n i n t r o d u c t i o n t o p r o b a b i l i t y
An event that has a probability close to 0 is unlikely to occur.
An event that has a probability close to 1 is likely to occur.
1
2
0
Impossible
1
Certain
Using these ideas, let us look at the three statements at the start of this section and express them in
the new language of probability.
Statement in terms of probability
It is certain that the sun will rise tomorrow.
The probability that the sun will rise tomorrow is 1.
If I toss a coin, getting a head and getting a tail are
equally likely.
If I toss a coin, the probability of getting a head is
1 and the probability of getting a tail is 1.
2
2
The probability of finding a plant that speaks
English is 0.
Example 1
PA
G
There is no chance of finding a plant that speaks
English.
ES
Original statement
FI
N
AL
A TV game show contestant is shown three closed doors and told that there is a prize behind
only one of the doors. If the contestant opens one of the doors, what is his probability of
winning the prize?
Solution
From the point of view of the quiz contestant the prize is equally likely to be behind each of
1
the doors. So the probability of the contestant winning the prize is .
3
430
I C E - E M M at h em at ic s y e a r 7
Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400
Exercise 17A
Example 1
1 Complete each of the following probability statements.
a It is certain that an iceblock will melt in the sun in Sydney.
The probability that an iceblock will melt in the sun in Sydney is _______
b There is no chance of finding a dog that speaks German.
The probability of finding a dog that speaks German is _______
ES
c If there are three red discs and three blue discs in a bag and you take one out without
looking in the bag, you are equally likely to get a red disc or a blue disc.
If there are three red discs and three blue discs in a bag and you take one out without
looking in the bag, the probability of getting a red disc is _______ and the probability of
getting a blue disc is _______
d If it is Thursday today, tomorrow will be Friday.
If it is Thursday today, the probability that it will be Friday tomorrow is _______
G
e If today is the 31st of January, there is no chance that tomorrow will be the 1st of May. If
today is the 31st of January, the probability that tomorrow is the 1st of May is _______
N
AL
17 B
PA
2 Make up some statements for which there is a corresponding probability statement
1
involving the probabilities 0, 1 or .
2
Experiments and counting
A standard die is a cube with each face marked with a number from 1 to 6 or with
marks as shown in the diagram.
FI
Suppose we roll a standard die. Since the outcomes 1, 2, ..., 6, are equally likely
(assuming that the die is fair; that is, it is not ‘loaded’), the probability of getting
1
1
a 1 is , the probability of getting a 2 is , and so on.
6
6
The total probability is 1
Question:
If we roll a die, what is the chance of getting 1 or 2 or 3 or 4 or 5 or 6?
Answer:
It is certain that we will get one of the numbers 1, 2, 3, 4, 5, 6, so we say that the
total probability is 1.
C h apt e r 1 7 P r o babi l it y
Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400
431
1 7 B E x pe r i m e n t s a n d c o u n t i n g
Sample space
Rolling a die and recording the face that shows up is an example of doing an experiment.
The numbers 1, 2, ..., 6, are called the outcomes of this experiment. The complete set of possible
outcomes for any experiment is called the sample space of that experiment. For example, we can
write down the sample space for the experiment of rolling a die as:
1 2 3 4 5 6
1
In this case, each outcome is equally likely and has probability .
6
Events
ES
An event is something that happens. In everyday life, we speak of sporting events, or we may
say that the school concert was a memorable event. We use the word ‘event’ in probability in a
similar way.
G
For example, suppose that we roll a die and we are interested in getting a prime number. In this case
‘the number is prime’ is the event that interests us. Some of the outcomes will give rise to this event.
For instance, if the outcome is 2, then the event ‘the number is prime’ has occurred. We say that the
outcome 2 is favourable to the event ‘the number is prime’. If the outcome is 4, then the event ‘the
number is prime’ does not occur. The outcome 4 is not favourable to the event.
Of the six outcomes, three – 2, 3 and 5 – are favourable to the event ‘the number is prime’. Three
outcomes – 1, 4 and 6 – are not favourable to the event ‘the number is prime’.
Example 2
PA
In many situations ‘success’ means favourable to the event and ‘failure’ means not favourable to the
event.
Hassan rolls a die. Hassan is hoping for a perfect square number.
N
AL
a What is the sample space?
b What are the outcomes that are favourable to the event ‘the result is a perfect square’?
Solution
FI
a The sample space is:
1 2 3 4 5 6
b The only perfect squares in the sample space are 1 and 4, so the outcomes favourable to
the event are:
1 4
Probability of an event
Suppose I roll a die. A natural question to ask is: What is the probability that I get a 1 or a 6?
To make sense of this, we have to decide how the question is to be interpreted.
There are six possible outcomes:
1 2 3 4 5 6
432
I C E - E M M at h em at ic s y e a r 7
Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400
1 7 B E x pe r i m e n t s a n d c o u n t i n g
All are equally likely. Of these, only two are favourable to the event ‘the result is 1 or 6’. They are:
1 6
2 1
= because all of the outcomes 1, 2, 3, 4, 5
6 3
and 6 are equally likely. In general, the situation is as follows.
We say that the probability of getting a 1 or a 6 is
Probability of an event
For an experiment in which all of the outcomes are equally likely:
number of outcomes favourable to that event
total number of outcomes
ES
Probability of an event =
Here is a good way of setting out the solution to a question about the probability of an event.
G
Example 3
PA
I roll a die. What is the probability that the result is a perfect square?
Solution
The sample space of all possible outcomes is:
1 2 3 4 5 6
N
AL
The outcomes that are favourable to the event ‘the result is a perfect square’ can be circled,
as shown.
1 2 3 4 5 6
FI
Probability that the result is a perfect square =
number of favourable outcomes
total number of outcomes
2
6
1
=
3
=
Notation
From now on, we will often write ‘P’ instead of ‘probability of’. Thus we write the previous
result as:
P(perfect square) =
1
3
C h apt e r 1 7 P r o babi l it y
Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400
433
1 7 B E x pe r i m e n t s a n d c o u n t i n g
Example 4
I roll a die. What is the probability of getting a 2 or a 3 or a 4 or a 5?
Solution
The sample space, with the favourable outcomes circled, is:
1 2 3 4 5 6
number of favourable outcomes
total number of outcomes
4
=
6
2
=
3
The word ‘not’
PA
P(2 or 3 or 4 or 5) = 1 − P(1 or 6). Why is this so?
G
Class discussion
ES
P(2 or 3 or 4 or 5) =
Sometimes we need to recognise the word ‘not’ hidden in the question. The next example gives a
simple illustration of what we mean by this.
N
AL
Example 5
Joe is playing a board game. If he tosses a 1 with the die, he goes to jail. What is the
probability that he does not go to jail?
Solution
FI
This is the same as the probability of not getting a 1, which means that Joe gets a 2, 3, 4,
5 or 6.
So P (not a 1) = P (2, 3, 4, 5 or 6)
5
=
6
Alternatively, we can consider the probability of getting anything except a 1.
Then P (not a 1) = 1 − P (1)
1
=1−
6
5
= , as before.
6
434
I C E - E M M at h em at ic s y e a r 7
Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400
1 7 B E x pe r i m e n t s a n d c o u n t i n g
The words ‘random’ and ‘randomly’
In probability, we frequently hear expressions such as ‘chosen randomly’ or ‘chosen at random’, as
in the following situations:
• A classroom contains 23 students. A teacher comes into the room and chooses a student at random
to answer a question about history.
What does this mean? It means that the teacher chose the student as if she knew nothing at all
about the students. Another way of interpreting this is to imagine that the teacher had her eyes
closed and had no idea who was in the class when she chose a student.
• In a TV quiz show, a prize wheel is spun. The contestant is asked to guess which number the
wheel will stop at.
ES
The results of the spins are random and the contestant has no idea where the wheel will stop.
Example 6
PA
Solution
G
In a pick-a-box show, there are five closed boxes, each containing one snooker ball. Two of
the boxes each contain yellow balls and the others contain a green ball, a black ball and a red
ball, respectively. The contestant will win a prize if she chooses a yellow ball. She chooses a
box at random and opens it. What is the probability that she wins a prize?
Here is the sample space with the favourable outcomes circled. They are equally likely,
because she chooses at random.
N
AL
Y Y G B R
2
P(prize) =
5
Example 7
A school has 1200 students. The table below gives information about whether or not each
student plays a musical instrument.
Girls
Plays a musical instrument
325
450
Does not play an instrument
225
200
FI
Boys
Each student has a school number between 1 and 1200.
Each student’s number is written on a card and the 1200 cards are placed in a hat. One
card is chosen randomly from the hat. What is the probability that the number pulled out
belongs to:
a
b
c
d
a boy?
a girl?
a student who does not play an instrument?
a boy who plays a musical instrument?
C h apt e r 1 7 P r o babi l it y
Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400
435
1 7 B E x pe r i m e n t s a n d c o u n t i n g
Solution
b Number of students = 1200
Number of girls = 450 + 200
= 650
650
P(girl’s number) =
1200
13
=
24
a Number of student = 1200
Number of boys = 325 + 225
= 550
550
P (boy’s number) =
1200
11
=
24
G
ES
c Number of students = 1200
Number who do not play a musical instrument = 225 + 200
= 425
425
P(student does not play a musical instrument) =
1200
17
=
48
d Number of students = 1200
N
AL
PA
Number of boys who play a musical instrument = 325
325
P(boy who plays a musical instrument) =
1200
13
=
48
Exercise 17B
Example 2
1 A bag contains 10 marbles numbered 1 to 10. A marble is taken out.
a Give the sample space for this experiment.
FI
b Write down the outcomes favourable to the event ‘a marble with an odd number
is taken out’.
2 A die is rolled. Write down the outcomes favourable to each given event.
a An even number is rolled.
b An odd number is rolled.
c A number divisible by 3 is rolled.
d A number greater than 1 is rolled.
436
3
b Write down the outcomes favourable to the event ‘the number
obtained is less than 4’.
1
4
2
3 aFor the spinner shown opposite, write down the sample space of
the experiment ‘spinning the spinner’.
5
I C E - E M M at h em at ic s y e a r 7
Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400
Example
3, 4
1 7 B E x pe r i m e n t s a n d c o u n t i n g
4 A die is rolled. Work out the probability that:
a an even number is rolled
b an odd number is rolled
c a number divisible by 3 is rolled
d a number greater than 1 is rolled
5 A bag contains 10 marbles numbered 1 to 10. If a marble is taken out at random, what is the
probability of getting:
a a 3?
b an even number?
c a number greater than 2?
d a number divisible by 3?
e a number greater than 1?
6 The spinner shown opposite is spun. Write down the probability of:
3
b obtaining an odd number
1
4
ES
2
a obtaining a 4
5
PA
G
7 The letter tiles for the word MELBOURNE are placed in a box.
One piece is withdrawn at random. What is the probability of obtaining:
a an M?
c a vowel?
d a consonant?
8 A die is rolled. What is the probability of:
N
AL
Example 5
b an E?
a obtaining a 3?
b not obtaining a 3?
c obtaining a number divisible by 3?
d obtaining a number not divisible by 3?
9 A bag contains three red marbles numbered 1 to 3, five green marbles numbered 4 to 8, and
two yellow marbles numbered 9 and 10. A single marble is withdrawn at random. Find the
probability that:
b the marble is green
c the marble is yellow
d the number is odd
e the number is greater than 6
f the number is green and even
FI
a the marble is numbered 3
10 The letters of the alphabet are written on flashcards and put into a bag. A card is drawn out
at random. Find the probability that the letter is:
a D
b not W, X, Y or Z
c a vowel
d in the word ‘PERTH’
e in the word ‘CANBERRA’
C h apt e r 1 7 P r o babi l it y
Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400
437
1 7 B E x pe r i m e n t s a n d c o u n t i n g
Example 6
11 There are three green apples and two red apples in a bowl. Georgia is blindfolded and
randomly chooses one of the apples. What is the probability that she chooses a green apple?
12 A bag of sweets contains 10 red ones, nine green ones, six yellow ones and five blue ones.
They are all the same size and shape. You reach in and take one out at random. Find the
probability that the sweet you pick is:
a blue
b green or yellow
d purple
e not yellow or blue
c not red
13 Cards with numbers 1–100 written on them are placed in a box. One card is randomly
chosen. What is the probability of obtaining a number divisible by 5?
Normal jelly beans
Double-flavoured jelly beans
Green
Red
250
400
175
75
ES
14 A bowl contains green and red normal jelly beans and green and red double-flavoured jelly
beans. The numbers of the different colours and types are given in the table below.
G
Example 7
A jelly bean is randomly taken out of the bowl. Find the probability that:
b it is a green jelly bean
PA
a it is a double-flavoured jelly bean
c it is a green normal jelly bean
e it is a red jelly bean
d it is a red double-flavoured jelly bean
15 In a traffic survey taken during a 30-minute period, the number of people in each passing
car was noted, and the results tabulated as follows.
1
2
3
4
5
Number of cars
60
50
40
10
5
N
AL
Number of people in car
What is the probability that:
a there was only one person in a car during this period?
b there was more than one person in a car during this period?
c there were fewer than four people in a car during this period?
FI
d there were five people in a car during this period?
16 A door prize is to be awarded randomly at the end of a concert. The numbers of people at
the concert in different age groups are given in the table below.
Age group
0−5
6−11
12−18
19−30
30−40
40+
Number of people in age group
10
150
350
420
125
85
What is the probability of the prize winner being:
438
a in the 0–5 age group?
b in the 19–30 age group?
c aged 40 or less?
d aged between 12 and 30?
e older than 5?
f older than 40?
I C E - E M M at h em at ic s y e a r 7
Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400
Review exercise
1 There are eight marbles, numbered 1 to 8, in a bowl. A marble is randomly taken out.
What is the probability of getting:
a an 8? b an even number? c a number less than 6? d a number divisible by 3?
2 Twelve cards are put into a hat. Five of the cards are red and are numbered 1 to 5, and
the other seven cards are blue and are numbered 6 to 12. A card is taken randomly from
the hat. Find the probability of taking out:
b a card with an even number
c a card with a prime number
d a red card
ES
a a blue card
e a card with a number greater than 13
a the P?
b a B?
G
3 The letter tile pieces for the letters of the word ‘PROBABILITY’ are put into a box. One
piece is chosen at random. What is the probability of choosing:
c a vowel?
d a consonant?
PA
4 Seven red marbles numbered 1 to 7 and six blue marbles numbered 8 to 13 are placed in
a box. A marble is taken randomly from the box. Find the probability of taking out:
a a red marble
b a blue marble
c a marble with a prime number
d a marble with a multiple of 3
e a red marble with an even number
f a blue marble with an odd number
N
AL
g a red marble with a number less than 6 on it
5 A large bowl contains two types of lollies: chocolates and toffees. The chocolates and
toffees are manufactured by the Yummy Sweet Company and the ACE Confectionary
Company. The number of chocolates and toffees in the bowl of the two different brands
are given in the table below.
ACE Confectionary Company
250
170
300
140
FI
Chocolates
Toffees
Yummy Sweet Company
A lolly is randomly taken out of the bowl. Find the probability that:
a it is a chocolate manufactured by the Yummy Sweet Company
b it is a toffee manufactured by the Ace Confectionary Company
c it is a chocolate
d it is a toffee
6 The 11 letter tiles for the letters of the word ‘PROSPECTIVE’ are put into a box.
One piece is chosen at random. What is the probability of choosing:
a the I?
b a P?
c an E?
d a vowel?
C h apt e r 1 7 P r o babi l it y
Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400
439
7 Each number from 1 to 500 is written on a card and put into a box. A card is withdrawn
at random. What is the probability of obtaining a card with:
a a number less than 100? b a number divisible by 10? c a number divisible by 5?
Challenge exercise
ES
1 In a raffle, 50 tickets are sold. The tickets are numbered from 1 to 50. They are placed in
a hat and one is drawn out at random to win a prize. Find the probability that the number
on the winning ticket:
b is not 28
c is even
d is less than or equal to 20
e is an even number less than 20
f contains the digit 7
h contains the digit 2 at least once
PA
g does not contain the digit 9
G
a is 43
i contains the digit 2 only once
j does not contain the digit 2
2 Suppose that the raffle tickets in Question 1 are coloured: those numbered 1 to 25 are
green, the tickets numbered 26 to 40 are yellow and the rest are blue. Find the probability
that the ticket that wins the prize:
a is blue
b is not green
d is both green and blue
e is even-numbered and blue
f is prime and yellow
N
AL
c is green or blue
3 Thomas, Leslie, Anthony, Tracie and Kim play a game in which there are three equal
prizes. No player can win more than one prize.
a List the 10 ways the prizes can be allocated.
b What is the probability that Leslie wins a prize?
FI
c What is the probability that Leslie does not win a prize?
4 Find the probability that a randomly chosen three-digit number is divisible by:
a three
b seven
5 A bag initially contains 15 red balls and 8 blue balls. Some of the red balls are removed.
1
A ball is then drawn out at random. The probability that it is a red ball is . How many
3
red balls were removed?
6 Describe how you can use a coin to decide between three menu items with equal
probability.
7 The six faces of a die are labelled with the numbers −3, −2, −1, 0, 1, and 2. The die
is rolled twice. What is the probability that the product of the two numbers is negative
(less than zero)?
440
I C E - E M M at h e matic s y e a r 7
Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400