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)

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’

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?

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.

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.

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.

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,

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,

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) =

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

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

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)

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.

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 –

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.