**Principles of Counting**

Theories of Counting

Definition of Outcomes

Outcomes

There are 5 possible outcomes

Possible outcomes......

What are the possible outcomes in this competition?

You are then in a javelin throwing competition with 4 people altogether in it

You could possibly come 1st, 2nd, 3rd or 4th!

Similar to the other event.....

The consequences of any event/experiment are called the outcomes

Imagine you are running in a race with 5 people in it, what are the possible outcomes?

You could come 1st, 2nd, 3rd, 4th or 5th

Can you think of an example of something that has outcomes??

Simple?

At the end of the sports day you have competed in two events.

List out the possible outcomes for both events....

You could:

1st in race, 1st in Javelin

1st in race, 2nd in Javelin

2nd in race, first in Javelin.....etc.

How many outcomes are there in total?

Multiply the number

of outcomes in each event.....

Event 1: Race - How many outcomes?

Event 2: Javelin - How many outcomes?

Conclusion:

At the end of the day there were

20 possible outcomes between

the race and the javelin

More examples

this way!!

Other events with outcomes:

When you throw a dice......

Possible outcomes?

You could throw a:

1, 2, 3, 4, 5, or a 6

What if you threw a dice, and straight away after you flipped a coin.....?

Event 1:

6 possible outcomes

Event 2:

2 possible outcomes

In total in this event there are

12 possible outcomes!

PAge 41,

Q1 and Q2

Possible outcomes:

10, 11, 12, 13, 14

There are 5 possible outcomes

Event 1: Bag A

3 possible outcomes

Event 2: Bag B

2 possible outcomes

Total possible outcomes

= 6

Event 1:

Main course

3 outcomes

Event 2:

Desert:

2 outcomes

Total possible outcomes:

= 6

Tree Diagram

Spin 1:

Green

Blue

Spin 2

Green

Blue

Green

Blue

Spin 3

Green

Blue

Green

Blue

Green

Blue

Green

Blue

Outcomes

Green, Green, Green

Green, Green, Blue

Blue, Blue, Blue

Blue, Blue, Green

Blue, Green, Green

Green, Blue, Blue

Green, Blue, Green

Blue, Green, Blue

Manchester United and Tottenham are to

play each other in three games during in the

season.

Use a tree diagram to list the possible outcome

over the three games

**V**

Tree Diagram

Game 1

Man U

Draw

Spurs

Man U

Draw

Spurs

Man U

Draw

Spurs

**Game 2**

Man U

Draw

Spurs

Man U

Draw

Spurs

Man U

Draw

Spurs

Man U

Draw

Spurs

Man U

Draw

Spurs

Man U

Draw

Spurs

Man U

Draw

Spurs

Man U

Draw

Spurs

Man U

Draw

Spurs

Man U

Draw

Spurs

Man U, Man U, Man U

Man U, Draw, Man U

Man U, Draw, Draw

Man U, Man U, Spurs

Man U, Man U, Draw

Man U, Draw, spurs

Man U, Spurs, Man U

Man U, Spurs, Spurs

Man U, Spurs, Draw

Draw, Man U, Man U

Draw, Man U, Draw

Draw, Man U, Spurs

Spurs, Man U, Spurs

Draw, Draw, Man U,

Draw, Draw, Spurs,

Draw, Draw, Draw

Draw, Spurs, Man U

Draw, Spurs, Draws

Draw, Spurs, Spurs

Spurs, Man U, Draw

Spurs, Man U, Man U

Spurs, Draw, Man U

Spurs, Draw, Draw

Spurs, Draw, Spurs

Spurs, Spurs, Man U

Spurs, Spurs, Spurs

Spurs, Spurs, Draw

Game 3

List of Outcomes

Two way tables

Two spinners are spun one after another. Spinner 1

has number 1-4 on it, while Spinner 2 has numbers

6 - 9 on it

(i) How many outcomes are there?

(ii) How many outcomes add up to 8?

(iii) How many outcomes add up to greater than 7?

How many add up to 8?

How many are greater than 7?

How many of the outcomes are greater than 8?

How many of the outcomes are less than 2?

Which is more likely, odd or even?

What fraction are even?

What % of the scores are odd?

**+**

**+**

**x**

How many ways off the

island are there??

There are 4 places to leave the island

How many ways onto the other

island are there?

There are 6 ways onto the island

6 x 4 = 24

There are 24 possible routes to get

from one island to the other

First digit:

9 options

2nd Digit

8 options

3rd digit

7 options

Total Codes:

9 x 8 x 7

= 504 codes

If you could use the same number each time?

1st digit:

9 choices

2nd digit:

9 choices

3rd digit:

9 choices

9 x 9 x 9

= 729 codes

Ball 1:

42 choices

Ball 5:

38 choices

Ball 3:

40 choices

Ball 2:

41 choices

Ball 4:

39 choices

Ball 6:

37 choices

Total possibilities: 42 x 41 x 40 x 39 x 38 x 37

= 3,776,965,920

What about the

European Lotto???

Euromillions: You need 5 numbers from a pot of 50 balls, AND 2 numbers from a pot of 11.....

So that's:

50 x 49 x 48

x 47 x 46

= 254,251,200

And....

11 x 10

= 110

= 254,251,200 x 110

= 27,967,632,000 possible combinations!!!

**Probablity**

Probablility

What comes to your mind

when you hear "probablity"

Importance of Maths????

**Probability Scale**

Impossible

Certain

**0**

**1**

**Terminology**

Trial - An experiment in Probabity

Outcome - The possible results of a trial

Sample Space - The set of all possible ___________outcomes from a trial

For Example..........

**A teacher is putting his 6 pupils in groups of two. He puts Michael, Elaine and Martin's names in one bag, and Peter, Megan and Sarah in another bag. He then draws them out randomly.**

Use a tree diagram to list out the sample space.

Use a tree diagram to list out the sample space.

Bag 1

Possible outcomes

A teacher is putting his 6 pupils in groups of two. He puts Michael, Elaine and Martin's names in one bag, and Peter, Megan and Sarah in another bag. He then draws them out randomly.

Use a tree diagram to list out the sample space.

Michael

Elaine

Martin

Peter

Megan

Sarah

Peter

Megan

Sarah

Peter

Megan

Sarah

Bag 2

Possible Outcomes

Sample Space

1. Michael, Peter

2. Michael, Megan

3. Michael, Sarah

4. Elaine, Peter

5. Elaine, Megan

6. Elaine, Sarah

7. Martin, Peter

8. Martin, Megan

9. Martin, Sarah

Sample Space

1. Michael, Peter

2. Michael, Megan

3. Michael, Sarah

4. Elaine, Peter

5. Elaine, Megan

6. Elaine, Sarah

7. Martin, Peter

8. Martin, Megan

9. Martin, Sarah

Questions:

How many possible groups are there altogether?

What is the probability that Michael and Sarah will be picked together first?

What is the probability that the first group picked will be all boys?

What is the probability that the first group picked will be all girls?

What is the probability that the first group picked will be a boy and a girl?

Christmas Exam

Christmas Exam

Q1. Indices, Scientific Notation (old book)

Q2. Simultaneous Eqs and Quadratics (old book)

Q3. Probability (Active Maths)

Q4. Coordinate Geometry (Active Maths)

Q5. Trigonometry (Active Maths)

What is the sample space?

1, 2, 3, 4, 5, 6

(i) 1/6

(ii) 1/6

(iii) 0

(iv) 3 even numbers...

3/6

= 1/2

(v) 2/6

= 1/3

(iii) 1

(certain)

What is the sample space here?

Ace - 10

Jack

Queen

King

13cards

Ace - 10

Jack

Queen

King

13cards

Ace - 10

Jack

Queen

King

13cards

Ace - 10

Jack

Queen

King

13cards

Page 67

How many students are there in the class?

20 students

(i) 7/20

(ii) 4/20

(iii) 11/20

(iv) 3/20

(v) 16/20

Sample Space

1, 3, 6, 8, 9, 77, 89

How many outcomes altogether?

= 7