ch4 - STAT 200 Chapter 4 Probability: The Study of...

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STAT 200 Chapter 4 Probability: The Study of Randomness Probability concepts (Section 4.2) A sample space S is the set of of all possible outcomes of a random phenomenon. An event is an outcome or a combination of outcomes from a random phenomenon. We denote an event by an uppercase letter, e.g., A, B, C. e.g., Rolling a die: rolling a ‘6’ is an event. Rolling a ‘5’ is another event. Rolling two ‘4”s in two rolls is also an event. The notation P ( A ) denotes the probability of an event A that will occur. Properties of P ( A ): 1. 0 P ( A ) 1 P ( A ) = 0 implies event A is impossible. P ( A ) = 1 implies event A is certain. The larger the P ( A ), the higher the chance that the event A will occur. 2. The sum of the probabilities of all the non-overlapping events in the sample space is equal to 1. The sample space contains all possible outcomes of a random phenomenon, so by the above property, P ( S ) = 1. Terminologies and probability rules (Sections 4.2 and 4.5) Some terminologies and notations: A or B (A and B are events) The notation “A or B” means either event A or event B or both occur. A and B (A and B are events) The notation “A and B” means event A and event B occur together. complement of an event The complement of an event A is the event that A does not occur. We denote the complement of A by A c . disjoint Two events A and B are said to be disjoint if the occurrence of A prevents the occurrence of B. In other words, the two events cannot occur together. independent Two events A and B are said to be independent if the occurrence of A does not alter the probability that B will occur. 1
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conditional probability of B given A (A and B are events) The notation P ( B | A ) denotes the probability of event B given that event A has occurred. P ( B | A ) = P ( A and B ) P ( A ) , P ( A ) > 0 If all the possible outcomes in a sample space are equally likely, then the probability of an event A is given by: P ( A ) = number of outcomes in event A total number of outcomes in the sample space Determining P ( A or B ) and P ( A and B ): 1. If events A and B are disjoint, then the probability that event A or event B or both will occur is P ( A or B ) = P ( A ) + P ( B ) [Addition Rule] Otherwise, P ( A or B ) = P ( A )+ P ( B ) - P ( A and B ) [General Addition Rule, Section 4.5] 2. If events A and B are independent, then the probability that event A and event B will occur together is P ( A and B ) = P ( A ) × P ( B ) [Multiplication Rule] Otherwise, P ( A and B ) 6 = P ( A ) × P ( B ) , and P ( A and B ) = P ( A ) × P ( B | A ) [General Multiplication Rule, Section 4.5] For a series of events A i ’s (1 , 2 ,...,m ), they are independent if and only if P ( A i and A j ) = P ( A i ) × P ( A j ) for all i 6 = j P ( A i and A j and A k ) = P ( A i ) × P ( A j ) × P ( A k ) for all i 6 = j 6 = k . . . = . . . P
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This note was uploaded on 01/17/2011 for the course STAT 200 taught by Professor Karim during the Spring '08 term at The University of British Columbia.

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ch4 - STAT 200 Chapter 4 Probability: The Study of...

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