Lecture Notes Chapter 2.pdf - CHAPTER 2 PROBABILITY 2.1 Sample Space(a if we are interested in the color facing upward on each of the two tosses A

Lecture Notes Chapter 2.pdf - CHAPTER 2 PROBABILITY 2.1...

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CHAPTER 2 PROBABILITY 2.1 Sample Space A probability model consists of the sample space and the way to assign probabilities. Sample space & sample point The sample space S , is the set of all possible outcomes of a statistical experiment. Each outcome in a sample space is called a sample point . It is also called an element or a member of the sample space. For example, there are only two outcomes for tossing a coin, and the sample space is S = { heads, tails } , or , S = { H, T } . If we toss a coin three times, then the sample space is S = { HHH, HHT, HTH, THH, HTT, TTH, THT, TTT } . E XAMPLE 2.1. Consider rolling a fair die twice and observing the dots facing up on each roll. What is the sample space? There are 36 possible outcomes in the sample space S , where S = ( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 ) ( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 ) ( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 ) ( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 ) ( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 ) ( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 ) N OTE . You may use a tree diagram to systematically list the sample points of the sample space. E XAMPLE 2.2. A fair six-sided die has 3 faces that are painted blue ( B ), 2 faces that are red ( R ) and 1 face that is green ( G ). We toss the die twice. List the complete sample space of all possible outcomes. (a) if we are interested in the color facing upward on each of the two tosses. (b) if the outcome of interest is the number of red we observe on the two tosses. N OTE . A statement or rule method will best describe a sample space with a large or infinity number of sample points. For example, if S is the set of all points ( x , y ) on the boundary or the interior of a unit circle, we write a rule/statement S = { ( x , y ) | x 2 + y 2 1 } . E XAMPLE 2.3. List the elements of each of the follow- ing sample spaces: (a) S = { x | x 2 - 3 x + 2 = 0 } (b) S = { x | e x < 0 } N OTE . The null set , or empty set, denoted by φ , con- tains no members/elements at all. 2.2 Events Event An event is a subset of a sample space. Refer to Example 2.1 . “the sum of the dots is 6” is an event. It is expressible of a set of elements E = { ( 1 , 5 ) ( 2 , 4 ) ( 3 , 3 ) ( 4 , 2 ) ( 5 , 1 ) } Complement The event that A does not occur, denoted as A 0 , is called the complement of event A . E XAMPLE 2.4. Refer to Example 2.1 . What are the complement events of
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4 Chapter 2. Probability (a) event A “the sum of the dots is greater than 3” (b) event B “the two dots are different” Intersection The intersection of two events A and B , denoted by A B , is the event containing all elements that are common to A and B . E XAMPLE 2.5. Refer to the preceding example. Give A B 0 , A 0 B and A 0 B 0 . Mutually Exclusive Events that have no outcomes in common are said to be disjoint or mutually exclusive . Clearly, A and B are mutually exclusive or disjoint if and only if A B is a null set.
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