Mathematics Q3 PDF

MATHEMATICS Q3 Talasalin | Miguel ​ Tree Diagram: display all the possible Topic Outline: outcomes of an event. Each branch in a tree ​ Topic I: Illustration of Permutations d...

MATHEMATICS Q3 Talasalin | Miguel ​ Tree Diagram: display all the possible Topic Outline: outcomes of an event. Each branch in a tree ​ Topic I: Illustration of Permutations diagram represents a possible outcome. ​ Topic II: Factorial Notations ​ Topic III: PERMUTATION: Derivation of formula ​ Topic IV: Conditional Permutation ​ Topic V: Permutation (Finding the UNKNOWN) ​ Topic VI: CIRCULAR PERMUTATION ​ Topic VII: COMBINATION OF OBJECTS ​ Topic VIII: Combination (Finding the UNKNOWN) ​ Topic VIIII: Probability of Simple Events ​ Topic X: Probability of Compound Events ​ Topic XI: UNION and INTERSECTIONof Events ​ Topic XII: MUTUALLY INCLUSIVE AND EXCLUSIVE EVENTS. ​ Table: displays all the possible TOPIC outcomes of an event in a tabular way. SUB-TOPIC SUPPORTING DETAILS Illustration of Permutations Permutation: is an arrangement of objects in which order is important. ORDER MATTERS. Examples: ​ Opening a combination lock of a luggage. ​ Possible number of plate number LTO can create ​ Decoding/Cracking Codes ​ Winning in a singing contest. ​ Fundamental Counting Principle: a rule ​ Determining the top three winners in a Math quiz used to count the total number of possible bee outcomes in a situation. It states that if there ​ Selecting a class president, vice president, and are n ways of doing something, and m ways a secretary. of doing another thing after that, then there ​ Seating arrangements are n x m ways to perform both of these actions. METHODS IN ILLUSTRATING AND DETERMINING THE PERMUTATION OF OBJECTS ​ Systematic Listing: involves the actual list of all possible outcomes in an organized and systematic way. Type your initials here | 1  MORE EXAMPLE FACTORIAL NOTATION MORE EXAMPLE 1.5! + 2! = (5 * 4 * 3 * 2 * 1) + (2 * 1) = 120 + 2 = 122 2.8! - 5! = (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) - (5 * 4 * 3 * 2 * 1) = 40 ,320-120 = 40 ,200= (24)(2) 3.4! 2! = (4 * 3 * 2 * 1)(2 * 1) = 48 Factorial Notations Factorial Notation: is the product of all positive integers less than or equal to n. It is denoted by n! 2  PERMUTATION ​ b. Permutation of All Objects Taken r at a (DERIVATION OF FORMULA) Time: The number of permutations of n objects taken r at a time is n! determined by A. PERMUTATION WHEN REPETITION IS ALLOWED ​ The number of permutations of n objects when repetition is allowed is determined by the formula: where: the formula: n is the number of (n-r)! Objects r is the number of objects selected. where: n is the number of objects r is the number of objects selected. Conditional Permutation B. PERMUTATION WHEN REPETITION IS NOT ALLOWED -​ It is when permutation situation is transformed into a problem imposing some ​ a. Permutation Of All Objects Taken All at a restrictions. Time: The number of permutations of n objects taken all at a time is determined by the formula: EXAMPLE where: n is the number of objects. 3  ​ In general, there is no first place in the arrangement of things in a circular pattern. So when each person moves his/her position by one place clockwise or counterclockwise, the relative position are not changed. Permutation (Finding the UNKNOWN) Find the value of n: FORMULA The number of permutation of n objects arranged in a circle is determined by the formula: P = (n-1)! where n is the number of objects. Find the value of r: Formula 2: The number of permutations of n different things CIRCULAR PERMUTATION around a key ring and the like is: P = (n - 1)! ​ There is a special case of permutation 2 where the arrangement of things is in a circular pattern. It is called circular permutation. The most common example of this type is the seating arrangement of people around a circular table. ​ When people are seated around a circular table, we cannot say that one is seated on the first seat unlike what we have seen on linear permutations (arrangement of things in a line). MORE EXAMPLE 4  COMBINATION OF OBJECTS Combination (Finding the UNKNOWN) A Combination: is an arrangement/selection of items in which Find the value of n: order does not matter. ORDER DOES NOT MATTER! To find the number of Combinations of in items chosen at a time, you can use the formula EXAMPLES Find the value of r: 5  PROPERTIES OF PROBABILITY: 1. A probability of 1 means the event is certain to happen. 2. A probability 0 means the event is impossible to happen. Note: Decreasing if ilan digit ng final answer ganon den dami ng numbers na ilalagay mo sa numerator(decrea Experiment - is a process of investigation from which results are observed or recorded Outcome - is the possible result of an experiment Probability of Simple Events Sample Space - is the set of all possible outcomes for a probability experiment or activity. It is usually denoted by ​ PROBABILITY is the branch of mathematics letter S. concerning events and numerical description of how Event - is the subset of all possible outcomes or sample space likely they are to occur of an experiment. ​ EXPERIMENT is a Scientific procedures undertaken to make a discovery especially in a laboratory ​ OUTCOME is the way a thing turns out; a consequences ​ EVENTS is a thing that happens especially one of importance; a gathering ​ SAMPLE SPACE is the random trial of results of that set. FORMULA Probability of Simple Events - refers to an event with a single outcome. Ex. Tossing a coin, rolling a die. The theoretical probability of an event (E) is defined as the ratio of the number of favorable outcome to the total number of outcome. PROBABILITY: field of a mathematics that deals with chance. Probability is a numerical value between 0 and 1. EXAMPLES: 6  Probability of Compound Events Compound Event ​ Is an event which has more than one possible outcomes. ​ It includes two or more simple events. Examples: a. Tossing 2 coins b. Rolling the dice 2 Types of Compound Events 1. Independent Events: involves two or more events in which the outcome of one event DOES NOT affect the outcome of any other events. Problems with replacement. Examples: 1. Rolling a die and getting a 6 and tossing a coin and getting a head 2. Two coins and getting two heads. ​ For joint probability, the multiplication rule is used. ​ If events A and B are independent events, then the joint probability is given by: P(A and B with replacement) = P(A) x P(B) EXAMPLES: 7 2. Dependent Events - involves two or more events in which The probability of union of events A and B, written P(AUB), the outcome of one event DOES affect the outcome of any is the set of all the elements that are in A, or in B, or in both A other events. Problem w/o replacement. and B. Examples: P(AUB) = P(A) + P(B) - P(A∩B) 1. Studying hard, getting a high grade. P(AUB) = P(A) + P(B) 2. Eating a lot, gaining weight. Where: FORMULA: P(A) is the probability of event A; P(A and B) = P(A) x P(B given A) P(B) is the probability of event B; and P(A∩B) is the probability of both events. Examples: -​ As you can see the denominator is decreasing, which means it cannot be repeated again (DEPENDENT) -​ CLUE; “without replacement”, so if may nakita ka na ganto then dependent gagamitin mong formula UNION and INTERSECTION of Events PROBABILITY OF AN INTERSECTION OF TWO EVENTS: -​ The probability of intersection of events A and B, written P(A ∩ B), is the event containing the elements that are in both A and B. More example Where: -​ the union of sets corresponds to the logical "or" and n(both A and B) - is the number of ways both events will occur, the intersection of sets corresponds to the logical n(S) - is the number of all possible outcomes in a sample "and" space. 8  1.​ Intersection MUTUALLY INCLUSIVE AND EXCLUSIVE EVENTS. -​ Formula: P(AUB) = P(A) + P(B) -​ Denoted/logically: And -​ DEFINE MUTUALLY INCLUSIVE AND EXCLUSIVE Examples: EVENTS. -​ DIFFERENTIATE INCLUSIVE AND EXCLUSIVE EVENTS. -​ SOLVE PROBLEMS INVOLVING INCLUSIVE AND EXCLUSIVE EVENTS. MUTUALLY EXCLUSIVE EVENTS -​ Events are mutually exclusive if they cannot happen at the same time. Example: 1. If someone is a man, they can't be a woman 2. If you win a game, you can't lose it at the same time. 3. If you throw a die and get a five, you can't get a four at the same time. 4. In a single throw of a die, it can only have one number shown at the top face. The numbers on the face are mutually exclusive events -​ If A and B are mutually exclusive events then the probability of A happening OR the probability of B happening is the sum of their probabilities. Formula: P(A or B) = P(A) + P(B) 2.​ Union -​ Formula: P(AUB) = P(A) + P(B) - P(A∩B) MUTUALLY INCLUSIVE EVENTS -​ Denoted/logically: OR Examples: -​ Events are mutually inclusive if they can happen at the same time. Example: 1. If you are a man you may also be under the age of 21. 2. If you throw an even number, you may also have thrown a six. 3. If you draw a rectangle, it may also be a square. Formula: P(AUB) = P(A) + P(B) – P(A∩B) 9