# PROBABILITY. Probability  The likelihood or chance of an event occurring  If an event is IMPOSSIBLE its probability is ZERO  If an event is CERTAIN.

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PROBABILITY

Probability  The likelihood or chance of an event occurring  If an event is IMPOSSIBLE its probability is ZERO  If an event is CERTAIN its probability is ONE  So all probabilities lie between 0 and 1  Probabilities can be represented as a fraction, decimal of percentages Probabilty 0 0.51 Impossibe Unlikely Equally LikelyLikely Certain

Experimental Probability

Theoretical Probability  Key Terms:  Experimentthrowing coindie  # possible Outcomes, n(S)26  Sample Space, SH,T 1,2,3,4,5,6  Event A(A subset S)getting Hgetting even #  Probability  The probability of an event A occurring is calculated as:  P(A)= #A/#Outcomes  Examples  One letter selected from excellent consonant  dice   Expectation  The expectation of an event A is the number of times the event A is expected to occur within n number of trials,  E(A)=n x P(A)   coin, 30 tosses expectd # tails  rainfall = 20%, expected # days in sept?

Sample Space Sample Space can be represented as:  List  Grid/Table  Two-Way Table  Venn Diagram  Tree Diagram

Sample Space

Dice 2\Dice 1123456 1234567 2345678 3456789 45678910 56789 11 6789101112

Sample Space 3) TWO- WAY TABLE: A survey of Grade 10 students at a small school returned the following results: A student is selected at random, find the probability that: a) It is a girl b) The student is not good at math c) It is a boy who is good at Math CategoryBoysGirls Good at Math1719 Not good at Math 812 25 31 56 25 20

Sample Space 4) VENN DIAGRAM: The Venn diagram below shows sports played by students in a class: A student is selected at random, find the probability that the student: a) plays basket ball b) plays basket ball and tennis

Sample Space 5) TREE DIAGRAM: Note: tree diagrams show outcomes and probabilities. The outcome is written at the end of each branch and the probability is written on each branch. Represent the following in tree diagrams: a) Two coins are tossed b) One marble is randomly selected from Bag A with 2 Black & 3 White marbles, then another is selected from Bag B with 5 Black & 2 Red marbles. c) The state allows each person to try for their pilot license a maximum of 3 times. The first time Mary goes the probability she passes is 45%, if she goes a second time the probability increases to 53% and on the third chance it increase to 58%.

Sample Space 5) TREE DIAGRAM: a) Answer:

Sample Space 5) TREE DIAGRAM: b) Answer:

Sample Space 5) TREE DIAGRAM: c) Answer:

Types of Events  EXHAUSTIVE EVENTS: a set of event are said to be Exhaustive if together they represent the Sample Space. i.e A,B,C,D are exhaustive if: P(A)+P(B)+P(C)+P(D) = 1 Eg Fair Dice: P(1)+P(2)+P(3)+P(4)+P(5)+P(6)=

Types of Events A’ A EVENT AA’ (COMPLEMENTARY EVENT) Getting a 6 on a die Getting at least a 2 on a die Getting the same result when a coin is tossed twice

Types of Events COMPOUND EVENTS:  EXCLUSIVE EVENTS: a set of event are said to be Exclusive (two events would be “Mutually Excusive”) if they cannot occur together. i.e they are disjoint sets  INDEPENDENT EVENTS: a set of event are said to be Independent if the occurrence of one DOES NOT affect the other.  DEPENDENT EVENTS: a set of event are said to be dependent if the occurrence of one DOES affect the other. A B

Types of Events EXCLUSIVE/ INDEPENDENT / DEPENDENT EVENTS  Which of the following pairs are mutually exclusive events? Event AEvent B Getting an A* in IGCSE Math Exam Getting an E in IGCSE Math Exam Leslie getting to school lateLeslie getting to school on time Abi waking up lateAbi getting to school on time Getting a Head on toss 1 of a coinGetting a Tail on toss 1 of a coin Getting a Head on toss 1 of a coinGetting a Tail on toss 2 of a coin  Which of the following pairs are dependent/independent events? Event AEvent B Getting a Head on toss 1 of a coinGetting a Tail on toss 2 of a coin Alvin studying for his exams Alvin doing well in his exams Racquel getting an A* in Math Racquel getting an A* in Art Abi waking up lateAbi getting to school on time Taking Additional MathTaking Higher Level Math

Probabilities of Compound Events When combining events, one event may or may not have an effect on the other, which may in turn affect related probabilities A B Type of Probability MeaningDiagramCalculation Conditional Probability of A given B is the probability that A occurs given that event B has occurred. This basically changes the sample space to B Probability that event A AND event B will occur together. Generally, AND = multiplication Probability that either event A OR event B (or both) will occur. Generally, OR = addition A B A B A B

Examples – Using “Complementary” Probability 1. The table below show grades of students is a Math Quiz Find the probability that a student selected at random scored at least 2 on the quiz (i)By Theoretical Probability(ii) By Complementary Grade12345 Frequency57101612

Examples – Using “Conditional” Probability 1. The table below show grades of students is a Math Quiz A student selected at random, Given that the student scored more than 3,find the probability that he/she scored 5 Grade12345 Frequency57101612

Examples- Conditional Probability a. b. B B’ M F 155 145 300 140 160

Examples – Using “OR” Probability 1. A fair die is rolled, find the probability of getting a 3 or a 5. (i)By Sample Space(ii) By OR rule

Examples – Using “AND” Probability 1. A fair die is rolled twice find the probability of getting a 5 and a 5. (i)By Sample Space(ii) By AND rule

Examples – Using “OR” /“AND” Probability 1. A fair die is rolled twice find the probability of getting a 3 and a 5. (i)By Sample Space(ii) By AND/OR rule

Mixed Examples 1. From a pack of playing cards, 1 card is selected. Find the probability of selecting: a) A queen or a king b) Heart or diamond c) A queen or a heart d) A queen given that at face card was selected e) A card that has a value of at least 3 (if face cards have a value of 10 and Ace has a value of 1)

Mixed Examples 2. From a pack of playing cards, 1 card is selected noted and replaced, then a 2 nd card is selected and noted. Find the probability of selecting: a) A queen and then a king b) A queen and a king c) Heart or diamond d) Two cards of same number e) Two different cards

 All the hungry-bellies began begging for free food (especially Leslie and Samantha). So we did not get to finish these questions  Please write out the remaining examples and leave space for us to discuss tomorrow

Mixed Examples 3. From a pack of playing cards, 1 card is selected noted, it is NOT replaced, then a 2 nd card is selected and noted. Find the probability of selecting: a) A queen and then a king b) A queen and a king c) Heart or diamond d) Two cards of same number e) Two cards with different numbers

Using Tree Diagrams

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