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Each probability assignment is given byEach probability assignment is given bydividing the frequency (number of days) bydividing the frequency (number of days) bythe total frequency (total number of days).the total frequency (total number of days).
Relative Frequency MethodRelative Frequency Method
4/404/404/404/40
ProbabilityProbabilityNumber ofNumber of
Polishers RentedPolishers RentedNumberNumberof Daysof Days
When economic conditions and a company’sWhen economic conditions and a company’s circumstances change rapidly it might becircumstances change rapidly it might be inappropriate to assign probabilities based solely oninappropriate to assign probabilities based solely on historical data.historical data. We can use any data available as well as ourWe can use any data available as well as our experience and intuition, but ultimately a probabilityexperience and intuition, but ultimately a probability value should express our value should express our degree of beliefdegree of belief that the that the experimental outcome will occur.experimental outcome will occur.
The best probability estimates often are obtained byThe best probability estimates often are obtained by combining the estimates from the classical or relativecombining the estimates from the classical or relative frequency approach with the subjective estimate.frequency approach with the subjective estimate.
Some Basic Relationships of ProbabilitySome Basic Relationships of Probability
There are some There are some basic probability relationshipsbasic probability relationships that thatcan be used to compute the probability of an eventcan be used to compute the probability of an eventwithout knowledge of all the sample point probabilities.without knowledge of all the sample point probabilities.
Complement of an EventComplement of an Event Complement of an EventComplement of an Event
Intersection of Two EventsIntersection of Two Events Intersection of Two EventsIntersection of Two Events
The complement of The complement of AA is denoted by is denoted by AAcc.. The complement of The complement of AA is denoted by is denoted by AAcc..
The The complementcomplement of event of event A A is defined to be the eventis defined to be the event consisting of all sample points that are not in consisting of all sample points that are not in A.A. The The complementcomplement of event of event A A is defined to be the eventis defined to be the event consisting of all sample points that are not in consisting of all sample points that are not in A.A.
The union of events The union of events AA and and BB is denoted by is denoted by AA BB The union of events The union of events AA and and BB is denoted by is denoted by AA BB
The The unionunion of events of events AA and and BB is the event containing is the event containing all sample points that are in all sample points that are in A A oror B B or both.or both. The The unionunion of events of events AA and and BB is the event containing is the event containing all sample points that are in all sample points that are in A A oror B B or both.or both.
Union of Two EventsUnion of Two Events
SampleSpace SSampleSpace SEvent Event AA Event Event BB
The intersection of events The intersection of events AA and and BB is denoted by is denoted by AA The intersection of events The intersection of events AA and and BB is denoted by is denoted by AA
The The intersectionintersection of events of events AA and and BB is the set of all is the set of all sample points that are in bothsample points that are in both A A and and BB.. The The intersectionintersection of events of events AA and and BB is the set of all is the set of all sample points that are in bothsample points that are in both A A and and BB..
SampleSpace SSampleSpace SEvent Event AA Event Event BB
Intersection of Two EventsIntersection of Two Events
The The addition lawaddition law provides a way to compute the provides a way to compute the probability of event probability of event A,A, or or B,B, or both or both AA and and B B occurring.occurring. The The addition lawaddition law provides a way to compute the provides a way to compute the probability of event probability of event A,A, or or B,B, or both or both AA and and B B occurring.occurring.
Addition LawAddition Law
The law is written as:The law is written as: The law is written as:The law is written as:
The probability of an event given that another eventThe probability of an event given that another event has occurred is called a has occurred is called a conditional probabilityconditional probability.. The probability of an event given that another eventThe probability of an event given that another event has occurred is called a has occurred is called a conditional probabilityconditional probability..
A conditional probability is computed as follows :A conditional probability is computed as follows : A conditional probability is computed as follows :A conditional probability is computed as follows :
The conditional probability of The conditional probability of AA given given BB is denoted is denoted by by PP((AA||BB).). The conditional probability of The conditional probability of AA given given BB is denoted is denoted by by PP((AA||BB).).
The The multiplication lawmultiplication law provides a way to compute the provides a way to compute the probability of the intersection of two events.probability of the intersection of two events. The The multiplication lawmultiplication law provides a way to compute the provides a way to compute the probability of the intersection of two events.probability of the intersection of two events.
The law is written as:The law is written as: The law is written as:The law is written as:
Often we begin probability analysis with initial orOften we begin probability analysis with initial or prior probabilitiesprior probabilities..
Then, from a sample, special report, or a productThen, from a sample, special report, or a product test we obtain some additional information.test we obtain some additional information. Given this information, we calculate revised orGiven this information, we calculate revised or posterior probabilitiesposterior probabilities..
Bayes’ theoremBayes’ theorem provides the means for revising the provides the means for revising the prior probabilities.prior probabilities.
To find the posterior probability that event To find the posterior probability that event AAii will will occur given that eventoccur given that event B B has occurred, we applyhas occurred, we apply Bayes’ theoremBayes’ theorem..
Bayes’ theorem is applicable when the events forBayes’ theorem is applicable when the events for which we want to compute posterior probabilitieswhich we want to compute posterior probabilities are mutually exclusive and their union is the entireare mutually exclusive and their union is the entire sample space.sample space.