PROBABILITY Probability The likelihood or chance of an
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. 5 0 Impossibe Unlikely Equally Likely 1 Likely Certain
Experimental Probability
Theoretical Probability Key Terms: � � Experiment # possible Outcomes, n(S) Sample Space, S 1, 2, 3, 4, 5, 6 Event A (A subset S) throwing coin 2 die 6 H, T getting H getting 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 can be represented as: List Grid/Table Two-Way Table Venn Diagram Tree Diagram
Sample Space
Sample Space
Sample Space Dice 2Dice 1 1 2 3 4 5 6 7 8 9 4 5 6 7 8 9 10 11 12
Sample Space TWO- WAY TABLE: A survey of Grade 10 students at a small school returned the following results: Category Boys Girls 3) Good at Math 17 19 25 Not good at Math 8 12 20 56 25 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 31
Sample Space VENN DIAGRAM: The Venn diagram below shows sports played by students in a class: 4) A student is selected at random, find the probability that the student: a) plays basket ball b) plays basket ball and tennis
Sample Space 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%. 5)
Sample Space 5) a) TREE DIAGRAM: Answer:
Sample Space 5) b) TREE DIAGRAM: Answer:
Sample Space 5) c) TREE DIAGRAM: 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 EVENT A Getting a 6 on a die Getting at least a 2 on a die Getting the same result when a coin is tossed twice A’ (COMPLEMENTARY EVENT) A’
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 A B 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.
Types of Events EXCLUSIVE/ INDEPENDENT / DEPENDENT EVENTS � Which of the following pairs are mutually exclusive events? Event A Getting an A* in IGCSE Math Exam Leslie getting to school late Abi waking up late Getting a Head on toss 1 of a coin � Event B Getting an E in IGCSE Math Exam Leslie getting to school on time Abi getting to school on time Getting a Tail on toss 1 of a coin Getting a Tail on toss 2 of a coin Which of the following pairs are dependent/independent events? Event A Getting a Head on toss 1 of a coin Alvin studying for his exams Racquel getting an A* in Math Abi waking up late Taking Additional Math Event B Getting a Tail on toss 2 of a coin Alvin doing well in his exams Racquel getting an A* in Art Abi getting to school on time Taking 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 Type of Probability Meaning 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 Diagram A B A B Calculation 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
Examples – Using “Complementary” Probability 1. The table below show grades of students is a Math Quiz Grade 1 2 3 4 5 Frequenc y 5 7 10 16 12 Find the probability that a student selected at random scored at least 2 on the quiz (i)By Theoretical Probability (ii) By Complementary
Examples – Using “Conditional” Probability 1. The table below show grades of students is a Math Quiz Grade 1 2 3 4 5 Frequenc y 5 7 10 16 12 A student selected at random, Given that the student scored more than 3, find the probability that he/she scored 5
Examples- Conditional Probability M B B’ a. b. F 140 155 160 145 300
Examples – Using “OR” Probability A fair die is rolled, find the probability of getting a 3 or a 5. (i)By Sample Space (ii) By OR rule 1.
Examples – Using “AND” Probability A fair die is rolled twice find the probability of getting a 5 and a 5. (i)By Sample Space (ii) By AND rule 1.
Examples – Using “OR” /“AND” Probability 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 1.
Mixed Examples 1. a) b) c) d) e) From a pack of playing cards, 1 card is selected. Find the probability of selecting: A queen or a king Heart or diamond A queen or a heart A queen given that at face card was selected 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. a) b) c) d) e) 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 queen and then a king A queen and a king Heart or diamond Two cards of same number 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. a) b) c) d) e) 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 queen and then a king A queen and a king Heart or diamond Two cards of same number Two cards with different numbers
Using Tree Diagrams
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