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 0.5 1 Impossibe Unlikely Equally Likely Likely Certain

Experimental Probability Relative Frequency is an estimate of probability 𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦= 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝐸𝑣𝑒𝑛𝑡 𝑇𝑜𝑡𝑎𝑙 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 Approaches theoretic probability as the number of trials increases Example Toss a coin 20 times an observe the relative frequency of getting tails.

Theoretical Probability Key Terms: Each EXPERIMENT has a given number of specific OUTCOMES which together make up the SAMPLE SPACE(S). The probability of an EVENT (A) occurring must be such that A is subset of S Experiment throwing coin die # possible Outcomes, n(S) 2 6 Sample Space, S H,T 1,2,3,4,5,6 Event A (A subset S) getting H getting even #

Theoretical Probability The probability of an event A occurring is calculated as: 𝑃 𝐴 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 (𝐴) 𝑇𝑜𝑡𝑎𝑙 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑃𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑂𝑢𝑡𝑐𝑜𝑚𝑒𝑠 = 𝑛(𝐴) 𝑛(𝑆) Examples A fair die is rolled find the probability of getting: a “6” a factor of 6 a factor of 60 a number less than 6 a number greater than 6 One letter is selected from “excellent”. Find the probability that it is: an “e” a consonant One card is selected from a deck of cards find the probability of selecting: a Queen a red card a red queen A B 1 6 4 6 = 2 3 6 6 =1 5 6 0 6 =0 3 9 = 1 3 6 9 = 2 3 4 52 = 1 13 26 53 = 1 2 2 52 = 1 26

Theoretical Probability Conditional Probability 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 𝑃 𝐴|𝐵 = 𝑛(𝐴𝑎𝑛𝑑 𝐵) 𝑛(𝑩) Examples A fair die is rolled find the probability of getting: a “6” given that it is an even number a factor of 6, given that it is a factor of 8 One letter is selected from “excellent”. Find the probability that it is: a “l” given it is a consonant an “e”, given the letter is in excel One card is selected from a deck of cards find the probability of selecting: a Queen , given it is a face card a red card given it is a queen a queen, given it is red card A B 1 3 1,2} 𝑓𝑟𝑜𝑚 (1,2,4} 2 3 2 6 = 1 3 {e,e,e} from {e,x,c,e,l,l,e} 3 7 4 12 = 1 3 2 4 = 1 2 4 26 = 2 13

Theoretical Probability 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) Examples A coin is tossed 30 times. How many time would you expect to get tails? The probability that Mr Bennett wears a blue shirt on a given day is 15%. Find the expected number of days in September that he will wear a blue shirt? 1 2 ×30 =15 𝑡𝑖𝑚𝑒𝑠 15%×30=4.5 ≈5 𝑑𝑎𝑦𝑠

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

Sample Space LIST: Bag A: 1 Black , 1 white . Bag B: 1 Black, 1 Red One marble is selected from each bag. Represent the sample space as a LIST Hence state the probability of choosing the same colours ANSWER: 𝑆= 𝐵𝐵,𝐵𝑅,𝑊𝐵,𝑊𝑅 𝑃 𝑠𝑎𝑚𝑒 𝑐𝑜𝑙𝑜𝑢𝑟𝑠 = 1 4

Sample Space i)GRID: Two fair dice are rolled and the numbers noted Represent the sample space on a GRID Hence state the probability of choosing the same numbers ANSWER: 𝑆= P 𝑠𝑎𝑚𝑒 #𝑠 = 6 36 = 1 6

Sample Space ANSWER: ii)TABLE: Two fair dice are rolled and the sum of the scores is recorded Represent the sample space in a TABLE Hence state the probability of getting an even sum ANSWER: 𝑆= P 𝑒𝑣𝑒𝑛 𝑠𝑢𝑚 = 18 36 = 1 2 Dice 2\Dice 1 1 2 3 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: A student is selected at random, find the probability that: it is a girl the student is not good at math it is a boy who is good at Math it is a girl, given the student is good at Math the student is good at Math, given that it is a girl Category Boys Girls Good at Math 17 19 Not good at Math 8 12 36 20 25 31 56 P 𝐺𝑖𝑟𝑙 = 31 56 P 𝑁𝑜𝑡 𝑔𝑜𝑜𝑑 @𝑀𝑎𝑡ℎ = 20 56 = 5 14 P 𝐵𝑜𝑦, 𝑔𝑜𝑜𝑑 @𝑀𝑎𝑡ℎ = 17 56 P 𝐺𝑖𝑟𝑙| 𝑔𝑜𝑜𝑑 @𝑀𝑎𝑡ℎ = 19 36 P 𝑔𝑜𝑜𝑑 @𝑀𝑎𝑡ℎ|𝐺𝑖𝑟𝑙 = 19 31

Sample Space 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: plays basket ball plays basket ball and tennis Plays basketball given that the student plays tennis P 𝐵𝑎𝑠𝑘𝑒𝑡𝑏𝑎𝑙𝑙 = 17 27 P 𝐵𝑎𝑠𝑘𝑒𝑡𝑏𝑎𝑙𝑙&𝑇𝑒𝑛𝑛𝑖𝑠 = 4 56 P 𝐵𝑎𝑠𝑘𝑒𝑡𝑏𝑎𝑙𝑙|𝑇𝑒𝑛𝑛𝑖𝑠 = 4 17

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: Two coins are tossed 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. 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 TREE DIAGRAM: Answer:

Sample Space TREE DIAGRAM: Answer:

Sample Space TREE DIAGRAM: Answer:

P(A)+P(B)+P(C)+P(D) = 1 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 COMPLEMENTARY EVENTS: two events are said to be complementary if one of them MUST occur. A’ , read as “A complement” is the event when A does not occur. A and A’ () are such that: P(A) + P(A’) = 1 State the complementary event for each of the following Eg Find the probability of not getting a 4 when a die is tossed P(4’) = Eg. Find the probability that a card selected at random form a deck of cards is not a queen. P(Q’)= A’ A EVENT A A’ (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 A Event B Getting an A* in IGCSE Math Exam Getting an E in IGCSE Math Exam Leslie getting to school late Leslie getting to school on time Abi waking up late Abi getting to school on time Getting a Head on toss 1 of a coin Getting a Tail on toss 1 of a coin Getting a Head on toss 1 of a coin Getting a Tail on toss 2 of a coin Which of the following pairs are dependent/independent events? Alvin studying for his exams Alvin doing well in his exams Racquel getting an A* in Math Racquel getting an A* in Art Taking Additional Math 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 Diagram Calculation AND 𝑷 𝑨 ∩𝑩 Probability that event A AND event B will occur together. Generally, AND = multiplication 𝑷 𝑨 𝒂𝒏𝒅 𝒕𝒉𝒆𝒏 𝑩 =𝑷 𝑨 ×𝑷 𝑨|𝑩 Note: For Exclusive Events: since they cannot occur together then, 𝑷 𝑨 ∩ 𝑩 =𝟎 For Independent: Events: since A is not affected by the occurrence of B 𝑷 𝑨 ∩ 𝑩 =𝑷 𝑨 ×𝑷 𝑩 OR 𝑷 𝑨 ∪𝑩 Probability that either event A OR event B (or both) will occur. OR = addition 𝑷 𝑨 ∪𝑩 =𝑷 𝑨 +𝑷 𝑩 −𝑷 𝑨∩𝑩 since such events are disjoint sets, 𝑷 𝑨 𝑶𝑹 𝑩 =𝑷 𝑨 +𝑷 𝑩 A B A B A B