Engineering Data Analysis Module 2: Probability PDF

10/19/2022 1 2 ❑ Probability ❑ The study of probability and statisti...

10/19/2022 1 2 ❑ Probability ❑ The study of probability and statistical inference aids in the translation ▪ It is an index which measures the chance or likelihood that an event of sample information into something conclusive or inconclusive about resulting from a statistical experiment will occur. the scientific system under study. ▪ It is expressed as the ratio of number of outcomes pertaining to the desired event to the total number of possible outcomes. ❑ It allows us to quantify the strength or “confidence” that we have in our Number of Outcomes conclusions. More importantly, the conclusion drawn about the scientific Probability = system being studied helps an individual in the decision-making Total Number of Possible Outcomes process. ❑ Statistical Experiment ▪ It is a process of using sample information to draw conclusions about certain characteristics of a population that is under study. Engineering Data Analysis by Dr. Robert G. de Luna, PECE 3 Engineering Data Analysis by Dr. Robert G. de Luna, PECE 4 3 4 1  10/19/2022 ❑ Experiment ▪ It is a process which generates an observation, or a measurement commonly referred to as raw data or information. ❑ An experiment is considered to be a random or chance process if: ▪ all possible outcomes are known before the experiment is performed. ▪ no particular outcome can be predicted with certainty before the performance of the trial or experiment. ❑ Classical examples include the rolling of a die and the tossing of a coin. Engineering Data Analysis by Dr. Robert G. de Luna, PECE 5 5 6 ❑ Sample Space (S) ❑ Three Ways of Defining Sample Space ▪ It is a list of all possible outcomes of a statistical experiment defined in such a ▪ Listing or Roster Method way that exactly one of the elements will occur. It is done by enumerating all the elements of the sample space. ❑ Sample Point (x) ▪ Tree Diagram Method ▪ It is also known as member or element in the sample space It uses a “tree” to express the sequence of events of an experiment in ▪ It refers to each possible outcome in the sample space. chronological order. ❑ If the sample space has a finite number of elements, we may list the ▪ Defining Property or Rule Method members separated by commas and enclosed in braces. Thus, the sample It is done by choosing a property or characteristic common to all sample space S, of possible outcomes when a coin is flipped, may be written as points and then using this common characteristic to define the sample space. S = {H, T} Engineering Data Analysis by Dr. Robert G. de Luna, PECE 7 Engineering Data Analysis by Dr. Robert G. de Luna, PECE 8 7 8 2  10/19/2022 ❑ Example 1: Consider the experiment of tossing a die. Determine the ❑ Example 2: An experiment consists sample space if we are interested in of flipping a coin and then flipping it a A. all the numbers that shows on the top face. second time if a head H occurs. If a tail T occurs on the first flip, then a die B. all even numbers that shows on the top face. is tossed once. Determine the sample C. all odd numbers that shows on the top face. space of this experiment. ▪ Solution: ▪ Solution: A. S = {1, 2, 3, 4, 5, 6} with 6 elements S = {HH, HT, T1, T2, T3, T4, T5, T6} B. S = {2, 4, 6} with 3 elements C. S = {1, 3, 5} with 3 elements with 8 elements Engineering Data Analysis by Dr. Robert G. de Luna, PECE 9 Engineering Data Analysis by Dr. Robert G. de Luna, PECE 10 9 10 ❑ Example 3: Suppose that three items ❑ Example 4: Determine the sample space for are selected at random from a manufacturing process. Each item is A. set of cities in the world with a population over 1 million. inspected and classified defective, D, B. set of all points (x, y) on the boundary or the interior of a circle of or nondefective, N. Determine the radius 2 units with center at the origin. sample space of this statistical experiment. ▪ Solution: ▪ Solution: Format: S = { x | x is (statement) } where | is read “such that” S = {DDD, DDN, DND, DNN, NDD, A. S = {x | x is a city with a population over 1 million} NDN, NND, NNN} B. S = {(x, y) | (x, y) is a set of all points satisying x2 + y2 ≤ 4} with 8 elements Engineering Data Analysis by Dr. Robert G. de Luna, PECE 11 Engineering Data Analysis by Dr. Robert G. de Luna, PECE 12 11 12 3  10/19/2022 ❑ Event ▪ It is a subset of a sample space. ❑ Simple Event ▪ It is an event which consists of only one outcome. ❑ Compound Event ▪ It is an event which consists of more than one outcome and may be decomposed into simple events. Engineering Data Analysis by Dr. Robert G. de Luna, PECE 14 13 14 ❑ Roll a die and observe the number appearing on the top face. ❑ The sample space, S, is given as S = {1, 2, 3, 4, 5, 6}. ❑ The following are some of the possible events: Event Event A event that an odd number appears E event that a “3” appears B event that an even number appears F event that a “4” appears C event that a “1” appears G event that a “5” appears D event that a “2” appears H event that a “6” appears ❑ Events A and B are known as compound events. ❑ Events C to H are simple events. Engineering Data Analysis by Dr. Robert G. de Luna, PECE 15 15 16 4