r3_probability - 13.42 Design Principles for Ocean Vehicles...

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13.42 Design Principles for Ocean Vehicles Reading # 13.42 Design Principles for Ocean Vehicles Prof. A.H. Techet Spring 2005 1. Overview of basic probability Empirically, probability can be defined as the number of favorable outcomes divided by the total number of outcomes, in other words, the chance that an event will occur. Formally, the probability, p of an event can be described as the normalized “area” of some event within an event space, S , that contains several outcomes (events), A , which can include i the null set, . The probability of the event space itself is equal to one, hence any other event has a probability ranging from zero (null space) to one (the whole space). Simple events are those which do not share any common area within an event space, i.e. they are non-overlapping, whereas composite events overlap (see figure 1). The probability that an event will be in the event space is one: pS () = 1. A 1 A 2 A 4 A 3 A 1 A 2 A 4 A 3 S S Simple Events A Composite Events A i i 1. Simple and composite events within event space, S . ©2004, 2005 A. H. Techet 1 Version 3.1, updated 2/14/2005
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13.42 Design Principles for Ocean Vehicles Reading # DEFINE (see figure 2 for graphical representation): UNION: The union of two regions defines an event that is either in A or in B or in both regions. INTERSECTION: The intersection of two regions defines an event must be in both A and B. COMPLEMENT: The complement A is everything in the event space that is not in A, i.e. A . A B A B B union; A B intersection; A A A' not A; A' 2. Union, Intersection, and Complement. ©2004, 2005 A. H. Techet 2 Version 3.1, updated 2/14/2005
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13.42 Design Principles for Ocean Vehicles Reading # 1.1. Mutually Exclusive Events are said to be mutually exclusive if they have no outcomes in common. These are also called disjoint events. EXAMPLE: One store carries six kinds of cookies. Three kinds are made by Nabisco and three by Keebler. The cookies made by Nabisco are not made by Keebler. Observe the next person who comes into the store to buy cookies. They choose one bag. It can only be made by either Nabisco OR Keebler thus the probability that they choose one made by either company is zero. These events are mutually exclusive. 0 p ( A B ) = Mutually Exclusive (1) AXIOMS: For any event A () (1) pA 0 (2) pS () = 1 (all events) 1 , A 3 L (3) If AA 2 ,, , A are a collection of n mutually exclusive events then: ( ( A 2 A A ) = n p A ) 1 3 n i L i = 1 Probability can be seen as the normalized Area of the event, A i . Since ( = 1 ) 1 (2) i i then the probability of the null set is zero: p ()1 ( ∅=− S ) = 0 . (3) This holds since the probability of the event space, S , is exactly one. ©2004, 2005 A. H. Techet 3 Version 3.1, updated 2/14/2005
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13.42 Design Principles for Ocean Vehicles Reading # If AB BA ( B ∩≠ 0 and A ∪=∪ A ) where A and ( B A ) are mutually exclusive, then B ) () + p B ( p ( A = pA () B ) (4) becomes ( ( p ( ) = A ) + A ) (5) since B is simply the union of the part of B in A with the part of B not in A: B = ( BA ) ( ) .
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.22 taught by Professor Alexandratechet during the Spring '05 term at MIT.

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r3_probability - 13.42 Design Principles for Ocean Vehicles...

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