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MIT 13 42 - Design Principles for Ocean Vehicles

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13.42 Design Principles for Ocean Vehicles Reading # ©2004, 2005 A. H. Techet 1 Version 3.1, updated 2/14/2005 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), iA , which can include 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: () 1pS=. A1A2A4A3Simple Events AiA1A2A4A3Composite Events AiSS 1. Simple and composite events within event space, S .13.42 Design Principles for Ocean Vehicles Reading # ©2004, 2005 A. H. Techet 2 Version 3.1, updated 2/14/2005 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′. ABAA BABAA BAA'not A; A'intersection;union; 2. Union, Intersection, and Complement.13.42 Design Principles for Ocean Vehicles Reading # ©2004, 2005 A. H. Techet 3 Version 3.1, updated 2/14/2005 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. ()0pA B Mutually Exclusive∩=→ (1) AXIOMS: For any event A (1) () 0pA≥ (2) () 1pS= (all events) (3) If 123 nAAA A,,,,L are a collection of mutually exclusive events then: 1231()()nniipA A A A pA=∪∪∪∪ =∑L Probability can be seen as the normalized Area of the event, iA . Since ()1 ()1iipA pA=−≤ (2) then the probability of the null set is zero: ()1 ()0ppS∅=− =. (3) This holds since the probability of the event space, S , is exactly one.13.42 Design Principles for Ocean Vehicles Reading # ©2004, 2005 A. H. Techet 4 Version 3.1, updated 2/14/2005 If 0AB∩≠ and ()ABA BA′∪=∪ ∩ where A and ()BA′∩ are mutually exclusive, then ( ) () () ( )pA B pA pB pA B∪= + − ∩ (4) becomes ()()( )pAB pA pBA′∪= + ∩ (5) since B is simply the union of the part of B in A with the part of B not in A: ()( )BBA BA′=∩∪∩. (6) These two parts are Mutually Exclusive thus we can sum their probabilities to get the probability of B. So () ( ) ( )pBpBApBA′=∩+ ∩ (7) Looking back to equation 5 we can substitute in for ( )pBA′∩ with ( ) ( )pBpAB−∩. Therefore, the probability of the event AB∪ is equal the probability of A plus the probability of event B minus the probability that AB∩, i.e. ( ) () () ( )pA B pA pB pA B∪= + − ∩. (8) Example 1: Toss a fair coin. Event A heads= and event Btails=. ( ) 05 ( ) 05pA pB=.; = . Example 2: If A, B, and C are the only three events in S and are mutually exclusive events, where13.42 Design Principles for Ocean Vehicles Reading # ©2004, 2005 A. H. Techet 5 Version 3.1, updated 2/14/2005 ( ) 49 100pA=/ and () 48100pB =/ then () 3100pC=/. Example 3: Roll a six-sided die. Six possible outcomes, ( ) 1 6ipA=/ . Probability of rolling an even number: ()12(2)(4)(6)peven p p p=/ = + + . 2. Conditional Probability Conditional probability is defined as the probability that a certain event will occur given that a composite event has also occurred. We write this conditional probability as ( )pAB| and say "probability of A given B". Given that a composite event, M (see figure 3), has happened what is the probability that event iA also happened? By stating that event M has happened we then have excluded all events that do not overlap with M as possible outcomes. The implication is that now the event space has “shrunk” from S to M. Therefore we must redefine the probabilities of the events such that ()1pM = and all other events have ( ) 0ipA= if 0iMA∩= but if 0iMA∩≠ (i.e. if iA has some overlap with M) then 0 ( ) 1ipA≤≤ . The greater the overlap, the higher the probability of the event.13.42 Design Principles for Ocean Vehicles Reading # ©2004, 2005 A. H. Techet 6 Version 3.1, updated 2/14/2005 A1A2A4A3SM (composite event) 3. Composite Event M. Thus for any two events A and B with ( ) 0pB> the CONDITIONAL PROBABILITY of A given B has occurred is defined as: ()()()pABpABpB∩|= (9) which is conveniently rewritten as ()()()pAB pABpB∩=|⋅ (10) and is commonly referred to as the Multiplication Rule and is often an easier form of equation 9. Example 1: A gas station is trying to determine what the average customer needs from their station. The have determined the probability that a customer will check only his/her oil level or only his/her tire pressure and also the probability they will check both. ()()002pchecktires pT==. ()()01p check oil p L==.13.42 Design Principles for Ocean Vehicles Reading # ©2004, 2005 A. H. Techet 7 Version 3.1, updated 2/14/2005 ()()()001p check both p B p T L==∩=. Their next step is to determine the probability that a person checks their oil given they also checked their tire pressure. (1) Choose a random customer and find the probability that a customer has checked his tires given he/she checked the oil: ()001() 01() 01pT LpT LpL∩.|===.. (11) (2) Choose a random customer and find the probability that a customer has checked his oil given he/she checked the tires: ()001() 005() 02pL TpLTpT∩.|===.. (12) Example 2: What is the probability that the outcome of


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MIT 13 42 - Design Principles for Ocean Vehicles

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