MATH112-WEEK 6-LESSON 10-Introduction to Probability.pdf

STATISTICAL ANALYSIS WITH SOFTWARE APPLICATION Introduction to Probability MODULE 2-WEEK 6-LESSON 10 Excellence and Relevance REMELYN L....

STATISTICAL ANALYSIS WITH SOFTWARE APPLICATION Introduction to Probability MODULE 2-WEEK 6-LESSON 10 Excellence and Relevance REMELYN L. ASAHID-CHENG  MATH112 Excellence and Relevance  Probability theory is a branch of mathematics concerned with the study of random phenomena. In life, we often deal with uncertainty and stochastic quantities, due to one of the reasons being incomplete observability — therefore, we most likely work with sampled data. Now, suppose we want to draw reliable conclusions about the behavior of a random variable, despite the fact that we only have limited data and we simply do not know the entire population. Hence, we need some kind of way to generalize from the sampled data to the population, or in other words — we need to estimate the true data-generating process. Excellence and Relevance Excellence and Relevance Introduction to Probability Inferential statistics involves taking a sample and using sample statistics to infer the values of population parameters. Laws of probability can often allow the researcher to assign a probability that the inference is correct. Excellence and Relevance Introduction to Probability Classical Method of Assigning Probabilities An experiment is a process that produces outcomes. An event is the outcome of an experiment. In the classical method, the probability of an individual event occurring is determined by the ratio of the number of items in a population that contain the event (ne) to the total number of items in the population (N). o Each outcome is equally likely ne P( E ) = N where N = total number of possible outcomes of an experiment ne = the number of outcomes in which the event occurs out of N outcomes Excellence and Relevance Introduction to Probability Classical Method of Assigning Probabilities Because ne can never be greater than N, the highest value of a probability is 1. The lowest probability, if none of the N possibilities has the desired characteristic, e, is 0 Thus 0 ≤ P(E) ≤ 1 Example: Machine A always produces 40% of products and always has a 10% defective rate o Thus the probability of that a product is defective and came from Machine A is 0.04 o A priori probability – the probability can be determined before the experiment takes place Excellence and Relevance Introduction to Probability Relative Frequency of Occurrence Probability of an event occurring is equal to the number of times the event has occurred in the past divided by the total number of opportunities for the event to have occurred Number of Times an Event Occurred g P( E ) = Total Number of Opportunities for the Event to Have Occurred Based on historical data; the past may or may not be a good predictor of the future Example: a company wants to know the probability that its inspectors will reject a shipment from a particular supplier. o They have received 90 shipments in the past and rejected 10 o Thus the probability that they will reject is 10 = 0.11 90 Excellence and Relevance Introduction to Probability Subjective Probability Based on the insights or feelings of the person determining the probability. Different individuals may (correctly or incorrectly) assign different numeric probabilities to the same event. Examples: o An experienced airline mechanic estimates the probability that a particular plane will have a certain type of defect. o A doctor assigns a probability to the expected life span of a patient with cancer. Excellence and Relevance Structure of Probability Experiment: a process that produces an outcome o Sampling every 200th bottle of cola and weighing it o Auditing every 10th account o Testing drugs on patients and recording outcomes Event: an outcome of an experiment o There are 10 bottles that are too full o There are 3 accounts with problems o Half of the patients improve Elementary event: event that cannot be decomposed or broken down into other events o Elementary events are denoted by lowercase letters o Suppose that the experiment is to roll a die o Elementary events are to roll a 1, a 2, a 3, etc. o In this case, there are six elementary events, e1, e2, etc. Excellence and Relevance Structure of Probability 1,2,3,4,5,6 o Rolling 2 dice: the sample space has 36 elementary events Excellence and Relevance Structure of Probability Unions and Intersections Set notation is the use of braces to group numbers o The union of sets X, Y is denoted 𝑋 ∪ 𝑌 An element is part of the union if it is in set X, set Y, or both “X or Y” o The intersection of sets X, Y is denoted 𝑋 ∩ 𝑌 An element is part of the intersection if it is in set X and set Y beginunder line endunder line “X and Y” Excellence and Relevance Structure of Probability Mutually Exclusive Events o Events with no common outcomes o Occurrence of one event precludes the occurrence of the other event o Example: if you toss a coin and get heads, you cannot get tails Excellence and Relevance Structure of Probability Independent Events o The occurrence or nonoccurrence of one event does not affect the occurrence or nonoccurrence of the other event(s) o The probability of someone wearing glasses is unlikely to affect the probability that the person likes milk o Many events are not independent The probability of carrying an umbrella changes when the weather forecast predicts rain If events are independent, then: P ( X  Y ) = P ( X ) , and P (Y  X ) = P (Y ) P ( X  Y ) is the probability that X occurs given that Y has occurred. Excellence and Relevance  In probability theory and logic, a set of events is jointly or collectively exhaustive if at least Structure of Probability one of the events must occur. For example, when rolling a six-sided die, the events 1, 2, 3, 4, 5, and 6 balls of a single outcome are collectively exhaustive, because they encompass the entire range of possible outcomes. Collectively Exhaustive Events o Contains all possible elementary events for an experiment o The sample space for an experiment can be described as mutually exclusive and collectively exhaustive Complementary Events o The elementary events of an experiment not in X comprise its complement o Complementary events are denoted X′, which is pronounced as “not X” For example, rolling a 5 or greater and rolling a 4 or less on a die are P ( X ) = 1− P ( X ) complementary events, because a roll is 5 or greater if and only if it is not 4 or less. Excellence and Relevance Structure of Probability Counting the Possibilities The mn Counting Rule: If an operation can be done m ways and a second operation can be done n ways, then there are mn ways for the two operations to occur in order o A cafeteria offers 5 salads, 4 meats, 8 vegetables, 3 breads, 4 desserts, and 3 drinks How many meals are available? g 5  4  8  3  4  3 = 5760 Sampling from a Population with Replacement: Sampling n items from a population of size N with replacement would provide (N)n begin underl ine end underl ine possibilities o Six lottery numbers are drawn from the digits 0 to 9, with replacement There are (10)6 = 1,000,000 possible outcomes Excellence and Relevance Structure of Probability Counting the Possibilities Combinations: Sampling from a Population without Replacement Sampling n items from a population of size N without replacement provides the following number of possibilities N N! C N n = =  n  n!( N − n)!   A small law firm has 16 employees and 3 are to be selected randomly to represent the company at the annual meeting of the American Bar Association How many combinations of employees are possible? Excellence and Relevance Some Theorems in Counting Sample Points 1. If an operation can be performed in n1 ways, and if each of these a second operation can be performed in n2 ways, then the two operations can be performed in n1 x n2 ways. Example: How many sample points are in the sample space when a pair of dice is thrown once? Solution: 1st die, n1 = 6 2nd die, n2 = 6 Thus, n1 x n2 = 6 x6 = 36 Excellence and Relevance Some Theorems in Counting Sample Points 2. If an operation can be performed in n1 ways, and if for each of these a second operation can be performed in n2 ways, then the sequence of k operations can be performed in n1 x n2 x ….nk ways. Example: A college freshman must take a science course, a computer science course and a mathematics course. If he may select any of 3 sciences, any of 4 computer science courses and any of 2 mathematics courses, how many ways can he arrange his program? Solution: n1 = 3, n2 = 4, n3 = 2 Thus, n = 3 x 4 x 2 = 24 Excellence and Relevance Some Theorems in Counting Sample Points Definition: A permutation of a set of distinct objects is an ordered arrangement of these objects. An ordered arrangement of r elements of a set is called an r-permutation. Example: Let S = {1,2,3}. The ordered arrangement 3,1,2 is a permutation of S. The ordered arrangement 3,2 is a 2-permutation of S. The number of r-permutations of a set with n elements is denoted by P(n,r). The 2-permutations of S = {1,2,3} are 1,2; 1,3; 2,1; 2,3; 3,1; and 3,2. Hence, P(3,2) = 6. Excellence and Relevance Some Theorems in Counting Sample Points 3. The number of permutations of n distinct objects of n distinct objects is n! The factorial function, with the symbol ! Means to multiply a series of descending natural numbers. Example: a. The number of permutations of the four letters a, b, c and d is 4! which is equal to 4 x 3 x 2 x 1 = 24. b. What is 6!? 6! = 6 X 5 X 4 X 3 X2 X 1 = 720. Excellence and Relevance Example: Suppose that a saleswoman has to visit eight different cities. She must begin her trip in a specified city, but she can visit the other seven cities in any order she wishes. How many possible orders can the saleswoman use when visiting these cities? Solution: Excellence and Relevance  Example: How many permutations of the letters ABCDEFGH contain the string ABC ? Solution: We solve this problem by counting the permutations of six objects, ABC, D, E, F, G, and H. Excellence and Relevance Some Theorems in Counting Sample Points 4. The number of permutations of n distinct objects taken r at a time is 𝒏! 𝒏 𝑷𝒓 = 𝒏−𝒓 ! Examples: a. Two lottery tickets are drawn from 20 for first and second prizes. Find the number of sample points in the space S. 𝟐𝟎! 𝟐𝟎! 𝟐𝟎∙𝟏𝟗∙𝟏𝟖! 𝟐𝟎𝑷𝟐 = 𝟐𝟎−𝟐 ! = 𝟏𝟖! = 𝟏𝟖! = 𝟑𝟖𝟎 b. How many ways can a local chapter of the Toastmasters Club schedule three speakers for three different meetings if they are all available on any of five possible dates. 𝟓! 𝟓! 𝟓∙𝟒∙𝟑∙𝟐! 𝟓𝑷𝟑 = 𝟓−𝟑 ! = 𝟐! = 𝟐! = 𝟔𝟎 Excellence and Relevance  Example: How many ways are there to select a first-prize winner, a second prize winner, and a third-prize winner from 100 different people who have entered a contest? Solution: Excellence and Relevance Some Theorems in Counting Sample Points Excellence and Relevance Some Theorems in Counting Sample Points Excellence and Relevance Some Theorems in Counting Sample Points Excellence and Relevance Combinations Definition: An r-combination of elements of a set is an unordered selection of r elements from the set. Thus, an r-combination is simply a subset of the set with r elements. The number of r-combinations of a set with n distinct elements is denoted by C(n, r). The notation is also used and is called a binomial coefficient. (We will see the notation again in the binomial theorem in Section 6.4.) Example: Let S be the set {a, b, c, d}. Then {a, c, d} is a 3- combination from S. It is the same as {d, c, a} since the order listed does not matter. C(4,2) = 6 because the 2-combinations of {a, b, c, d} are the six subsets {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, and {c, d}. Excellence and Relevance Some Theorems in Counting Sample Points Excellence and Relevance Combinations 3 Example: How many poker hands of five cards can be dealt from a standard deck of 52 cards? Also, how many ways are there to select 47 cards from a deck of 52 cards? Solution: Since the order in which the cards are dealt does not matter, the number of five card hands is: 52! C ( 52,5 ) = 5!47! 52  51 50  49  48 = = 26 17 10  49 12 = 2,598,960 5  4  3  2 1 The different ways to select 47 cards from 52 is 52! C ( 52, 47 ) = = C ( 52,5 ) = 2,598,960 47!5! This is a special case of a general result. → Excellence and Relevance Combinations Example: How many ways are there to select five players from a 10-member tennis team to make a trip to a match at another school. Solution: By Theorem 2, the number of combinations is 10! C (10,5 ) = = 252. 5!5! Example: A group of 30 people have been trained as astronauts to go on the first mission to Mars. How many ways are there to select a crew of six people to go on this mission? Solution: By Theorem 2, the number of possible crews is 30! 30  29  28  27  26  25 C ( 30, 6 ) = = = 593, 775 6!24! 6  5  4  3  2 1 Excellence and Relevance EXAMPLES OF COMPUTING PROBABILITIES Excellence and Relevance EXAMPLES OF COMPUTING PROBABILITIES Excellence and Relevance EXAMPLES OF COMPUTING PROBABILITIES Excellence and Relevance EXAMPLES OF COMPUTING PROBABILITIES Excellence and Relevance EXAMPLES OF COMPUTING PROBABILITIES Excellence and Relevance  End of LESSON 10 Excellence and Relevance REMELYN L. ASAHID-CHENG