Download yea 9 Probability

Contents
D4 Probability
DD4.1 The language of probability
2
DD4.2 The probability scale
2
DD4.3 Calculating probability
2
DD4.4 Probability diagrams
2
DD4.5 Experimental probability
2
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The language of probability
Probability is a measurement of the chance
or likelihood of an event happening.
Describe the chance of drawing a red marble.
Impossible
‫مستحيل‬
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Unlikely
‫غير مرجح‬
even
Chance
‫متساوي الفرصة‬
Likely
‫مرجح‬
Certain
‫مؤكد‬
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The probability scale
The chance of an event happening can be shown on a
probability scale.
Meeting
with King
Henry VIII
A day of the
week starting
with a T
The next baby
born being a
boy
Getting a
number > 2
when roll a
fair dice
A square
having four
right angles
impossible
unlikely
even chance
likely
certain
Less likely
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More likely
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Fair games
A game is played with marbles in a bag.
One of the following bags is chosen for the game. The
teacher then pulls a marble at random from the chosen bag:
bag a
bag b
bag c
If a red marble is pulled out of the bag, the girls get a point.
If a blue marble is pulled out of the bag, the boys get a point.
Which would be the fair bag to use?
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Fair games
A game is fair if all the players have
an equal chance of winning.
Which of the following games are fair?
A dice is thrown. If it lands on a prime number team A
gets a point, if it doesn’t team B gets a point.
There are three prime numbers (2, 3 and 5) and three
non-prime numbers (1, 4 and 6).
Yes, this game is fair.
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Fair games
Nine cards numbered 1 to 9 are used and a card is drawn
at random.
If a multiple of 3 is drawn team A gets a point.
If a square number is drawn team B gets a point.
If any other number is drawn team C gets a point.
There are three multiples of 3 (3, 6 and 9).
There are three square numbers (1, 4 and 9).
There are four numbers that are neither square nor
multiples of 3 (2, 5, 7 and 8).
No, this game is not fair. Team C is more likely to win.
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Fair games
A spinner has five equal sectors numbered 1 to 5.
The spinner is spun many times.
5
1
If the spinner stops on an even
number team A gets 3 points.
4
2
If the spinner stops on an odd
3
number team B gets 2 points.
Suppose the spinner is spun 50 times.
We would expect the spinner to stop on an even number 20
times and on an odd number 30 times.
Team A would score 20 × 3 points = 60 points
Team B would score 30 × 2 points = 60 points
Yes, this game is fair.
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Bags of counters
Choose a blue counter
and win a prize!
bag a
bag b
bag c
You are only allowed to choose one counter at random from
one of the bags.
Which of the bags is most likely to win a prize?
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The probability scale
The chance of an event happening can be shown on a
probability scale.
Meeting
with King
Henry VIII
A day of the
week starting
with a T
The next baby
born being a
boy
Getting a
number > 2
when roll a
fair dice
A square
having four
right angles
impossible
unlikely
even chance
likely
certain
Less likely
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More likely
© Boardworks Ltd 2004
The probability scale
We measure probability on a scale from 0 to 1.
If an event is impossible or has no probability of occurring
then it has a probability of 0.
If an event is certain it has a probability of 1.
This can be shown on the probability scale as:
0
impossible
½
even chance
1
certain
Probabilities are written as fractions, decimal and, less often,
as percentages between 0 and 1.
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The probability scale
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Contents
D4 Probability
DD4.1 The language of probability
2
DD4.2 The probability scale
2
DD4.3 Calculating probability
2
DD4.4 Probability diagrams
2
DD4.5 Experimental probability
2
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Higher or lower
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Listing possible outcomes
When you roll a fair dice you are equally likely to get one
of six possible outcomes:
1
6
1
6
1
6
1
6
1
6
1
6
Since each number on the dice is equally likely the
probability of getting any one of the numbers is 1 divided
1
by 6 or
.
6
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Calculating probability
What is the probability of the following events?
1) A coin landing tails up?
P(tails) =
1
2
2) This spinner stopping on
the red section?
1
P(red) =
4
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3) Drawing a seven of hearts
from a pack of 52 cards?
P(7 of
)=
1
52
4) A baby being born on a
Friday?
P(Friday) =
1
7
© Boardworks Ltd 2004
Calculating probability
If the outcomes of an event are equally likely then we can
calculate the probability using the formula:
Probability of an event =
Number of successful outcomes
Total number of possible outcomes
For example, a bag contains 1 yellow,
3 green, 4 blue and 2 red marbles.
What is the probability of pulling a green
marble from the bag without looking?
P(green) =
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3
10
or 0.3 or 30%
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Calculating probability
This spinner has 8 equal divisions:
What is the probability of the
spinner landing on
a) a red sector?
b) a blue sector?
c) a green sector?
2
1
=
8
4
1
b) P(blue) =
8
4
1
c) P(green) =
=
8
2
a) P(red) =
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Calculating probability
A fair dice is thrown. What is the probability of getting
a) a 2?
b) a multiple of 3?
c) an odd number?
d) a prime number?
e) a number bigger than 6?
f) an integer?
a) P(2) = 1
6
2
1
b) P(a multiple of 3) =
=
6
3
3
1
c) P(an odd number) =
=
6
2
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Calculating probability
A fair dice is thrown. What is the probability of getting
a) a 2?
b) a multiple of 3?
c) an odd number?
d) a prime number?
e) a number bigger than 6?
f) an integer?
d) P(a prime number) =
3
1
=
6
2
Don’t write
0
6
e) P(a number bigger than 6) = 0
6
f) P(an integer) =
6
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= 1
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Calculating probabilities
Answer these questions giving each answer
as a fraction or 0 or 1.
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The probability of an event not occurring
The following spinner is spun once:
What is the probability of it landing on the yellow sector?
1
P(yellow) =
4
What is the probability of it not landing on the yellow sector?
3
P(not yellow) =
4
If the probability of an event occurring is p then the
probability of it not occurring is 1 – p.
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The probability of an event not occurring
The probability of a factory component being faulty is 0.03.
What is the probability of a randomly chosen component not
being faulty?
P(not faulty) = 1 – 0.03 = 0.97
The probability of pulling a picture card out of a full deck of
cards is
3
13
.
What is the probability of not pulling out a picture card?
3
10
P(not a picture card) = 1 –
=
13
13
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The probability of an event not occurring
The following table shows the probabilities of 4 events.
For each one work out the probability of the event not
occurring.
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Event
Probability of the
event occurring
Probability of the
event not occurring
A
3
5
2
5
B
0.77
0.23
C
9
20
11
20
D
8%
92%
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The probability of an event not occurring
There are 60 sweets in a bag.
10 are cola bottles,
20 are hearts,
1
are fried eggs,
4
the rest are teddies.
What is the probability that a sweet chosen at random
from the bag is:
5
a) Not a cola bottle P(not a cola bottle) =
6
b) Not a teddy
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45
3
P(not a teddy) =
=
60
4
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Adding mutually exclusive outcomes
If two outcomes are mutually exclusive then their probabilities
can be added together to find their combined probability.
For example, a game is played with the following cards:
What is the probability that a card is a moon or a sun?
1
1
and
P(sun) =
3
3
Drawing a moon and drawing a sun are mutually exclusive
outcomes so,
1
1
2
P(moon or sun) = P(moon) + P(sun) =
+
=
3
3
3
P(moon) =
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Adding mutually exclusive outcomes
What is the probability that a card is yellow or a star?
1
1
P(yellow card) =
and
P(star) =
3
3
Drawing a yellow card and drawing a star are not mutually
exclusive outcomes because a card could be yellow and a star.
P (yellow card or star) cannot be found simply by adding.
We have to subtract the probability of getting a yellow star.
P(yellow card or star) =
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1
1
1
3+3–1
5
+
–
=
=
3
3
9
9
9
© Boardworks Ltd 2004
The sum of all mutually exclusive outcomes
The sum of all mutually exclusive outcomes is 1.
For example, a bag contains red counters, blue counters,
yellow counters and green counters.
P(blue) = 0.15
P(yellow) = 0.4
P(green) = 0.35
What is the probability of drawing a red
counter from the bag?
P(blue or yellow or green) = 0.15 + 0.4 + 0.35 = 0.9
P(red) = 1 – 0.9 = 0.1
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Finding all possible outcomes of two events
Two coins are thrown.
What is the probability of getting two heads?
Before we can work out the probability of getting two heads
we need to work out the total number of equally likely
outcomes.
There are three ways to do this:
1) We can list them systematically.
Using H for heads and T for tails, the possible outcomes
are:
TH and HT are separate
TT, TH, HT, HH.
equally likely outcomes.
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Finding all possible outcomes of two events
2) We can use a two-way table.
Second coin
First
coin
H
T
H
HH
TH
T
HT
TT
From the table we see that there are four possible outcomes
one of which is two heads so,
1
P(HH) =
4
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Finding all possible outcomes of two events
3) We can use a probability tree diagram.
Outcomes
Second coin
First coin
H
HH
T
H
HT
TH
T
TT
H
T
Again we see that there are four possible outcomes so,
1
P(HH) =
4
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Finding the sample space
A red dice and a blue dice are thrown and their scores
are added together.
What is the probability of getting a total of 8 from both dice?
There are several ways to get a total of 8 by adding the
scores from two dice.
We could get a 2 and a 6, a 3 and a 5, a 4 and a 4,
a 5 and a 3, or a 6 and a 2.
To find the set of all possible outcomes, the sample
space, we can use a two-way table.
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Finding the sample space
+
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2
3
4
5
6
7
3
4
5
6
7
8
4
5
6
7
8
9
5
6
7
8
9
10
6
7
8
9
10
11
7
8
9
10
11
12
From the sample
space we can see
that there are 36
possible outcomes
when two dice are
thrown.
Five of these have
a total of 8.
5
P(8) =
36
© Boardworks Ltd 2004
Contents
D4 Probability
DD4.1 The language of probability
2
DD4.2 The probability scale
2
DD4.3 Calculating probability
2
DD4.4 Probability diagrams
2
DD4.5 Experimental probability
2
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Estimating probabilities based on data
Suppose 1000 people were asked whether they were leftor right-handed.
Of the 1000 people asked 87 said that they were lefthanded.
From this we can estimate the probability of someone being
left-handed as
87
1000
or 0.087.
If we repeated the survey with a different sample the results
would probably be slightly different.
The more people we asked, however, the more accurate our
estimate of the probability would be.
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Relative frequency
The probability of an event based on data from an
experiment or survey is called the relative frequency.
Relative frequency is calculated using the formula:
Number of successful trials
Relative frequency =
Total number of trials
For example, Ben wants to estimate the probability that a
piece of toast will land butter-side-down.
He drops a piece of toast 100 times and observes that it
lands butter-side-down 65 times.
65
13
Relative frequency =
=
100
20
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Relative frequency
Sita wants to know if her dice is fair. She throws it 200 times
and records her results in a table:
Number Frequency Relative frequency
1
31
2
27
3
38
4
30
5
42
6
32
31
200
27
200
38
200
30
200
42
200
32
200
= 0.155
= 0.135
= 0.190
= 0.150
= 0.210
= 0.160
Is the dice fair?
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Experimental probability
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Expected frequency
The theoretical probability of an event is its calculated
probability based on equally likely outcomes.
If the theoretical probability of an event can be calculated,
then when we do an experiment we can work out the
expected frequency.
Expected frequency = theoretical probability × number of trials
If you rolled a dice 300 times, how many
times would you expect to get a 5?
The theoretical probability of getting a 5 is
1
So, expected frequency =
× 300 = 50
6
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1
6
.
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Expected frequency
If you tossed a coin 250 times how many
times would you expect to get a tail?
1
Expected frequency =
× 250 = 125
2
If you rolled a fair dice 150 times
how many times would you expect
to a number greater than 2?
2
Expected frequency =
× 150 = 100
3
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Spinners experiment
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Worksheet ( 1 )
* Write the sample space ( all possible results ) when rolling a fair dice
‫اكتب فضاء العينة ( مجموعة جميع النواتج الممكنة ) عند إلقاء حجر نرد منتظم‬
Use the words : impossible , unlikely , even
chance , likely , certain to describe the
following events :
Zayed althani school
Math department
Mohamad badawi : [email protected]
‫ غير‬، ‫ مستحيل‬: ‫استخدم المصطلحات‬
‫ مؤكد‬، ‫ مرجح‬، ‫ متساوي الفرصة‬، ‫مرجح‬
: ‫لتصف األحداث التالية‬
1) The upper face is a number greater than 5. ……………….
2) The upper face is a prime number. ……………….
‫ على الوجه العلوي‬5 ‫ظهور عدد اكبر من‬
‫ظهور عدد أولي على الوجه العلوي‬
Coin 2
* You toss 2 coins together
H
‫ألقيت قطعتي نقود معا‬
H
1) Complete the table to show all possible results . ‫أكمل الجدول لتبين جميع النواتج الممكنة‬
Use the words : impossible , unlikely , even
chance , likely , certain to describe the
following events :
1) You will get 2 heads ……………….
2) At least one head ……………….
‫ غير‬، ‫ مستحيل‬: ‫استخدم المصطلحات‬
، ‫ مرجح‬، ‫ متساوي الفرصة‬، ‫مرجح‬
: ‫مؤكد لتصف األحداث التالية‬
‫ستحصل على صورتين‬
‫ستظهر صورة واحدة على األقل‬
3) You will get one tail exactly………………. ‫ستظهر الكتابة مرة واحدة بالضبط‬
Coin
1
T
T
Worksheet ( 2 )
Zayed althani school
Math department
2 cards were randomly drawn from a deck of 52 cards
Mohamad badawi : [email protected]
‫ ورقة‬52 ‫تم سحب ورقتين من علبة لعب الورق ( الشدة ) التي تحوي‬
1) Complete the table to show all possible results .
Use the words : impossible , unlikely , even
chance , likely , certain to describe the
following events :
‫أكمل الجدول لتبين جميع النواتج الممكنة‬
‫ غير‬، ‫ مستحيل‬: ‫استخدم المصطلحات‬
، ‫ مرجح‬، ‫ متساوي الفرصة‬، ‫مرجح‬
: ‫مؤكد لتصف األحداث التالية‬
Card 2
1) The 2 cards are of the same color. ……………….
‫البطاقتان من نفس اللون‬
2) The 2 cards are spades. ……………….
‫البطاقتان من نوع البستوني‬
Card
1
3) One of the cards was green. ……………….
‫احد البطاقتين خضراء‬
: Spade ‫بستوني‬
4) The 2 cards are either red or black or red and black……………….
‫البطافتان إما حمراوتان أو سوداوتان أو حمراء وسوداء‬
5) At least one of the 2 cards wasn’t a picture. ……………….
‫على األقل احدهما ليست صورة‬
: Clubs
‫سباتي‬
: Diamond ‫ديناري‬
: Heart ) ‫كبه ( قلب‬