Chapter 2 Probability - MAT 226 Fa 24 PDF

Dr. Re-Mi Hage MAT 226 Fa 24 Chapter 2 Introduction to probability Recall: A standard deck of 52 playing cards can be described as follows: Hearts (red) Ace 2 3 4 5 6 7 8 9 10 Jack Queen King Clubs (black) Ace 2 3 4 5 6...

Dr. Re-Mi Hage MAT 226 Fa 24 Chapter 2 Introduction to probability Recall: A standard deck of 52 playing cards can be described as follows: Hearts (red) Ace 2 3 4 5 6 7 8 9 10 Jack Queen King Clubs (black) Ace 2 3 4 5 6 7 8 9 10 Jack Queen King Diamonds (red) Ace 2 3 4 5 6 7 8 9 10 Jack Queen King Spades (black) Ace 2 3 4 5 6 7 8 9 10 Jack Queen King Cards labeled Ace, Jack, Queen, or King are called face cards. 2.1 Introduction to probability The key terms related to probability are: Sample Space (S): The set of all possible outcomes of a statistical experiment. Observations: Recorded data from an experiment, which can be numerical or categorical. Statistical Experiment: A process that generates data, where outcomes are subject to chance. Outcomes: The results of an experiment, which can vary due to uncertainty. Data Types: o Numerical Data: Quantitative measurements or counts. o Categorical Data: Classification based on criteria (e.g., defective/non-defective). Types of Studies: o Designed Experiments: Controlled tests to observe specific outcomes. o Observational Studies: Data collected without manipulation of variables. o Retrospective Studies: Analysis of existing historical data. Tree Diagram: A systematic way to list the elements of a sample space for clarity. Venn diagram can be used to present the sample space and events. Elements of Sample Space: Individual outcomes within the sample space, known as sample points. Event: collection of sample point or it’s a subset of a sample space. 1|Page Dr. Re-Mi Hage MAT 226 Fa 24 Example: Experiment Outcome Sample space Event Tossing a coin H or T S={H,T} Observing a head thus T: tail, H: head the event is A={H} Tossing a die 1 or 2 or 3 or 4 or 5 or If we are interested in Observing a number 6 the number that shows greater than 3 thus on the top face, the A={4,5,6} sample space is S={1,2,3,4,5,6} If we are interested only in whether the number is even or odd, the sample space is simply S2 = {even, odd} Example: Create a tree diagram that lists all the sequences of heads and tails obtained by tossing a coin three times. Example: S={1,2,3,4,5,6,7,8,9,10} A={1,2,3,4,5,6} B={5,6,7,8} C={8,9} 2|Page Dr. Re-Mi Hage MAT 226 Fa 24 2.2 Set Theory Let S be the universal set and A, B be two subsets of S. 1. Complements The complement of an event A with respect to S is the subset of all elements of S that are not in A. We denote the complement of A by 𝐴̅ = 𝐴𝑐 = 𝐴′ The absolute complement of A is the set: 𝐴̅ = 𝐴𝑐 = {𝑥 ∈ 𝑆|𝑥 ∉ 𝐴} Example : Let R be the event that a red card is selected from an ordinary deck of 52 playing cards, and let S be the entire deck. Then R’ is the event that the card selected from the deck is not a red card but a black card Consider the sample space S = {book, cell phone, mp3, paper, stationery, laptop}. Let A = {book, stationery, laptop, paper}. Then the complement of A is A’= {cell phone, mp3} The relative complement of A with respect to B is the set: 𝐵 − 𝐴 = {𝑥 ∈ 𝑆|𝑥 ∈ 𝐵 𝑎𝑛𝑑 𝑥 ∉ 𝐴} 3|Page Dr. Re-Mi Hage MAT 226 Fa 24 2. Union: The union of the two events A and B, denoted by the symbol A∪B, is the event containing all the elements that belong to A or B or both The union of A and B is the set 𝐴 ∪ 𝐵 = {𝑥|𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵} Example: Let A = {a, b, c} and B = {b, c, d, e}; then A ∪ B = {a, b, c, d, e} If M = {x | 3