1. Probability Lesson 1 Aims:
• To have a general understanding of probability.
• To be able to use a sample space diagram and a two way table.
• To be able to know and be able to use the addition law of probability.
P(A B) = P(A) P(B) – P(A B)
• To know probabilities add up to 1. (Complimentary Events)
• To know what mutually exclusive events are. P(A B) = P(A) + P(B)
• To know what exhaustive events are.
• To be able to use probability notation.
• To know and be able to use the independent rule:
P(A B) = P(A) × P(B)

2. Introduction to probability and notation
How likely am I to live to 100?
Will Coronation Street ever end?
Uncertainty is a feature of everyday life. Probability is an area of maths that addresses how likely things are to happen.
A statistics experiment will have a number of different o__________. The set of all possible outcomes is called the sample s___________ of the experiment.
Example If a normal dice is thrown the sample space would be
{ }.
An event is a collection of some of the outcomes from an experiment. For example, getting an even number on the dice.
If we let A be the event of getting an even number then we could write the probability this happens as P(A) =
The probability that A does not occur is denoted P(A′) and in this case is _____.
So,
P(A′) = 1 – P(A)
A is said to compliment A’

3. Introduction to probability
When two experiments are combined, the set of possible outcomes can be shown in a sample space diagram.
Example: A dice is thrown twice and the scores obtained are added together. Find the probability that the total score is 6. 6 7 8 9 10 11 12 5 4 3 2 1
There are 36 equally likely outcomes.
5 of the outcomes result in a total of 6.
Second throw
P(total = 6) =
First throw
This notation means “probability that the total = 6”.
Note: if events are said to be e______________, then their probabilities will total one.
E.g. Name a three events that make up an exhaustive set?

4. The outcomes that satisfy event A can be represented by a circle.
Venn diagrams
V______ diagrams can be used to represent probabilities.
The outcomes that satisfy event A can be represented by a circle.
The outcomes that satisfy event B can be represented by another circle. A B
The circles can be overlapped to represent outcomes that satisfy both events.

5. Addition properties
Two events A and B are called m__________ exclusive if they cannot occur at the same time.
For example, if a card is picked at random from a standard pack of 52 cards, the events “the card is a club” and “the card is a diamond” are mutually exclusive.
However the events “the card is a club” and “the card is a queen” are not mutually exclusive.
If A and B are mutually exclusive, then:
A B
In Venn diagrams representing mutually exclusive events, the circles do not overlap.
This symbol means ‘union’ or ‘OR’
Reads A or B or both.

6. Addition properties P(A B) = P(A) + P(B) – P(A B)
This addition rule for finding P(A B) is not true when A and B are not mutually exclusive.
The more general rule for finding P(A B) is:
P(A B) = P(A) P(B) – P(A B)
This symbol means ‘intersect’ or ‘AND’
Venn diagrams can help you to visualize probability calculations.

7. Addition properties This represents the other 3 queens.
Example: A card is picked at random from a pack of cards. Find the probability that it is either a club or a queen or both.
Card is a club = event C Card is a queen = event Q
This represents the other 3 queens.
This area represents the 12 clubs that are not queens.
This represents the queen of clubs.
So,

8. On w/b
Example: A card is picked at random from a pack of cards. Find the probability that it is either a heart or a picture card or both. A picture card is a Jack, Queen or King
So,

9. Addition properties
Example 2: If P(A′ B′) = 0.1, P(A) = 0.45 and P(B) = 0.75, find P(A B).
P(A′ B′) is the unshaded area in the Venn Diagram.
A B
We can deduce that:
P(A B) =
Using the formula, P(A B) = P(A) + P(B) – P(A B), we get:
So, P(A B) =

10. Two Way Tables
Exam Question: One player is chosen at random from the 30 players.
Find the probability that the chosen player is:
(i) British; (iv) a defender or a midfielder;
(ii) a goalkeeper; (v) British or an attacker but not both.
(iii) British and a goalkeeper;
Goalkeepers
Defenders
Midfielders
Attackers
British 2 4 5
Other 7 3

11. On w/b Exam Question: One pet is chosen at random from 40.
Find the probability that the chosen pet is:
(i) Female; (iv) a dog or a rabbit;
(ii) a cat; (v) Male or a Cat but not both.
(iii) Male and a snake;
Dog
Cat
Rabbit
Snake
Male 7 8 3 2
Female 10 1
Do exercise 2A page 39 questions 1, 2, 3 and 7.
Do exercise 2B page 47 questions 1, 2 and 5 only

12. Independent events
Two events are said to be independent if the occurrence of one has n___ e_________ on the probability of the second occurring.
For example, if a coin and a die are both thrown, then the events “the coin shows a head” and “the die shows an odd number” are independent events.
If A and B are independent, then:
P(A B) = P(A) × P(B)

13. Independent events
Example: A and B are independent events. P(A) = 0.7 and P(B) = 0.4.
a) Find P(A B)
b) Find P(A′ B).
A B
a) As A and B are independent,
b) P(A′ B) is the shaded region in the Venn diagram.
Next we’ll look at Tree diagrams which are sometimes a useful way of finding probabilities that involve a succession of events.

14. Independent events G G B G G B B G G B B G B B
Example: A bag contains 6 green and 4 blue counters. A counter is chosen at random from the bag and then replaced three times. Find the probability that the 3 counters chosen are:
a) all green b) not all the same colour.
0.6 G G
0.6
0.4 B G
0.6 G
0.6
0.4 B B
0.4
0.6 G
0.6 G
0.4 B
0.4 B
0.6
0.4 G B
a) P(GGG) =
0.4 B
1 –

15. On w/b
Example: A bag contains 6 green and 4 blue counters. A counter is chosen at random from the bag three times without replacement. Find the probability that the 3 counters chosen are:
a) all green b) not all the same colour. G G B G G
6/10 B B G G B B G B
a) P(GGG) = B
1 –
Do worksheet. Then ex 2C page 52.
Full solutions are on Moodle if you get stuck.