CalcProb

# CalcProb - Outline Sample Spaces and Events Probability and...

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Outline Sample Spaces and Events Probability and its Calculus Deﬁnition and Calculus of Probability Michael Akritas Michael Akritas Deﬁnition and Calculus of Probability

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Outline Sample Spaces and Events Probability and its Calculus Sample Spaces and Events Events and Set Operations Probability and its Calculus The Deﬁnition and Meaning of Probability Axioms and Calculus of Probability Michael Akritas Deﬁnition and Calculus of Probability
Outline Sample Spaces and Events Probability and its Calculus Events and Set Operations Deﬁnition The set of all possible outcomes of a random experiment is called the sample space of the experiment, and will be denoted by S . 1. Select a tin plate from the population of 500 tin plates, and let the outcome be the number of scratches on the plate. Then S = { 0 , 1 , 2 } . 2. Roll two dice. What is S ? 3. Count the number of insect damaged leaves in a plant having N leaves. 4. Observe the reaction time to a stimulus. What is S ? Michael Akritas Deﬁnition and Calculus of Probability

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Outline Sample Spaces and Events Probability and its Calculus Events and Set Operations In experiments with many possible outcomes, we may be interested in the probability that one of a certain collection of outcomes will occur. For example, the statement ” at most 3 damaged leaves includes the four outcomes of 0, 1, 2, or 3 damaged leaves. I Event is a subset of the sample space (i.e. a collection of individual outcomes) I We say that a particular event A has occurred if the outcome of the experiment is a member of (i.e. contained in) A . I The sample space of an experiment is an event which always occurs when the experiment is performed. Michael Akritas Deﬁnition and Calculus of Probability
Outline Sample Spaces and Events Probability and its Calculus Events and Set Operations Because events are sets, we give a quick review of the basic set operations. I The union , A B , of events A and B , is the event consisting of all outcomes that are either in A or in B or in both. I The intersection , A B , of A and B , is the event consisting of all outcomes that are in both A and B . I The complement , A 0 or A c , of A is the event consisting of all outcomes that are not in A . I The events A and B are said to be mutually exclusive disjoint if they have no outcomes in common. That is, if A B = , where denotes the empty set. Michael Akritas Deﬁnition and Calculus of Probability

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Outline Sample Spaces and Events Probability and its Calculus Events and Set Operations I Commutative Laws: a) A B = B A , b) A B = B A I Associative Laws: a) ( A B ) C = A ( B C ) b) ( A B ) C = A ( B C ) I Distributive Laws: a) ( A B ) C = ( A C ) ( B C ), b) ( A B ) C = ( A C ) ( B C ) I De Morgan’s Laws: ( A B ) c = A c B c b) ( A B ) c = A c B c Michael Akritas Deﬁnition and Calculus of Probability
Outline Sample Spaces and Events Probability and its Calculus Events and Set Operations Venn Diagrams The set operations are illustrated by Venn diagrams : Michael Akritas Deﬁnition and Calculus of Probability

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CalcProb - Outline Sample Spaces and Events Probability and...

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