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Unformatted text preview: Discussion 2 Emily Mower January 27, 2010 Topics to be covered: • Mutually exclusive vs. independent (Section 1) • Symmetries (Section 2) • Light bulb problem (Section 3) 1 Mutually exclusive vs. independent 1.1 Overview of the differences Two events are mutually exclusive if the occurrance of one event implies that the other event did not occur. For example, let’s say that event A is, “It rained on Friday,” and event B is, “it did not rain on Friday.” One can observe that if event A transpires, event B cannot. Thus, events A and B are mutually exclusive. Two events are independent if the occurrance of one event has no effect on the outcome of the other event. For example, let’s say that event A is, “My computer broke at work today,” and event B is, “It rained in France.” Clearly, the rain in France did not cause my computer to break and my computer breaking did not cause it to rain in France. Therefore, these two events are independent. 1 1.2 Unions and intersections Given: sets A, B Then: The union of sets A and B is: A ∪ B Given: sets A, B, C Then: The union of the three sets is: A ∪ B ∪ C Extend this to nsets A i The union of nsets is: S n i =1 A i = A 1 ∪ A 2 ∪ ... ∪ A n Given: sets A, B Then: The intersection of sets A and B is: A ∩ B Given: sets A, B, C Then: The intersection of the three sets is: A ∩ B ∩ C Extend this to nsets A i The intersection of nsets is: T n i =1 A i = A 1 ∩ A 2 ∩ ... ∩ A n Intuitively: • S n i =1 A i = the event that at least one of the A i occurred • T n i =1 A i = the event that all of the A i occurred 1.3 Elementary Properties of Probability 1.4 Useful properties • P ( A ) = 1 P ( A )) • P (Ø) = 0 • P ( A ) ≤ P ( B ) if A ⊂ B • P ( A ) ≤ 1 • P ( A ∪ B ) = P ( A ) + P ( B ) P ( A ∩ B ) 2 1.5 Conditional Probability Given: events A and B Mathematical definition: P ( B  A ) Definition (English): “The probability of B occurring, given A has occurred” The conditional probability of B, given A, denoted by P ( B  A ) is defined by: P ( B  A ) = P ( A ∩ B ) P ( A ) , provided P ( A ) > 0. 1.6 Mutually exclusive events Given: events A and B Sets A and B are mutually exclusive iff (if and only if) the sets have no items in common (i.e., A ∩ B = Ø) 1.7 Independent events Given: events A and B Events A and B are statistically independent iff: P ( A ∩ B ) = P ( A ) P ( B ) This can also be formulated using conditional probability: Events A and B are statistically independent iff:...
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This note was uploaded on 02/26/2010 for the course EE ? taught by Professor Mendel during the Spring '10 term at USC.
 Spring '10
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