Ch2_3021_Instructor.pdf - STAT 3021 Chapter 2 Probability...

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STAT 3021 Chapter 2. Probability 2.1, 2.2 Sample Space and Events Definition 1. The set of all possible outcomes of a statistical experiment is called the sample space and is represented by the symbol S . Example 1. a. Rolling a die and observe the number it shows on the top face. Then S = { 1 , 2 , 3 , 4 , 5 , 6 } b. Rolling two dice and observe the two numbers they show. Then S = { (1 , 1) , (1 , 2) , (1 , 3) , ... (6 , 5) , (6 , 6) } c. Rolling two dice and calculate the sum of the two numbers. Then S = { 2 , 3 , 4 , 5 , 6 , ..., 12 } Example 2. Suppose that we randomly select a student from class and ask a question. In the below situation, describe the sample space for the experiment. a. How much time (in hours) did the student spend studying during the last 24 hours? S = [0 , 20] b. In what state was the student born, given that he/she was born in the U.S.? S = { AL, AK, AZ, ..., WY } c. How many friends does the student have? S = { 0 , 1 , ... } In some experiment it is helpful to list the elements of the sample space systemically by means of a tree diagram. Example 3. An experiment consists of flipping a coin and then flipping it a second time if a head occurs. If a tail occurs on the first flip, then a die is tossed once. To list the elements of the sample space providing the most information, we construct the tree diagram below. Hence S = { HH, HT, T 1 , T 2 , T 3 , T 4 , T 5 , T 6 } 1
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STAT 3021 Sample space with a large or infinite number of sample points are best described by a statement or rule method. Example 4. 1. S = { x | x is a city with a population over 1 millions } = { New York City, Los Angeles, Chicago, .. } 2. S = { x | number of tosses needed to observe a head on a fair coin } = { 1 , 2 , ... } Definition 2. An event is a subset of a sample space. Example 5. Suppose we toss a fair coin 3 times and observe the results. 1. What is the sample space? S = { HHH, HHT, HTH, THH, HTT, THT, TTH, TTT } 2. Let A be the even that we observe exactly 2 heads on the 3 tosses. Write down all the elements in A . A = { HHT, HTH, THH } 3. Let B be the event that at least one tail is flipped. Write down all the elements in B . B = { HHT, HTH, THH, HTT, THT, TTH, TTT } Definition 3. The complement of an event A with respect to S is the subset of all elements of S that is not in A . We denote the complement of A by the symbol A 0 . Definition 4. The intersection of two events A and B , denoted by the symbol A B , is the event containing all elements that are common to A and B . Definition 5. Two events A and B are mutually exclusive or disjoint , if A B = , that is, if A and B have no elements in common. Definition 6. The union of the two events A and B , denoted by the symbol A B , is the event containing all the elements that belongs to A or B or both. Example 6. Suppose that you roll a die and observe the result. Let A be the event where we observe a multiple of 3. Let B be the event where we observe an even number. Let C be the event we observe 1.
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