Probability-solutions

Probability-solutions - Introduction to probability Note...

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Unformatted text preview: Introduction to probability Note that a formal/rigorous introduction to probability is beyond the scope of this course. Probability is a numerical measure of uncertainty. Uncertainties are abundant. Consider uncertainties about the weather, a medical diagnosis, success of a marriage, yield of a corn crop, the consequences of any decision etc. Examples: • A weather forecaster asserts that there is a 60% chance of snow today • A physician asserts that there is a 95% chance that an operation will be successfull • An engineer claims that a 6-sided die will land on “3” with probability 1/6. We will use the basic concepts of probability theory, in conjunction with knowledge about a population or process, in order to specify a model for observations from that population or process. 1 Experiments, outcomes, and events 1.1 Definitions Definition: experiment action or process that generates outcomes. Only one outcome can occur and we are usually uncertain which outcome this will be. Definition: sample space set of all outcomes of an experiment. Denoted by S . We will now discuss sample spaces for which we can enumerate the outcomes. Example 1.1: Specify the sample space for the following experiments 1. Toss a coin: S = { H,T } 2. Toss two coins: S = { HH,HT,TH,TT } 3. Toss a coin until the first “heads”: S = { H,TH,TTH,TTTH,... } 4. Roll one 6-sided die: S = { 1 , 2 , 3 , 4 , 5 , 6 } 1 5. Roll two 6-sided dice: S = { (1 , 1) , (1 , 2) ,..., (6 , 6) } Definition: event any subset of outcomes (any subset of S ) • an event occurs if the occurring outcome is in the event. • an event may have zero, one, many, or all outcomes of the experiment. Example 1.2: Experiment: test if two batteries work Let W = “works” and let F = “fails”. S = { WW,WF,FW,FF } Specify the outcomes in the following events: 1. Event A : “the first battery works” A = { WW,WF } 2. Event B : “at least one works” B = { WW,WF,FW } 3. Event C : “exactly one works” C = { WF,FW } 4. Event D : “both batteries fail” D = { FF } 1.2 Events are sets For any experiment with sample space S and events A and B . Definition: Union, A ∪ B the event consisting of all outcomes in either A or B or both. Note that A ∪ B = B ∪ A . Definition: Intersection, A ∩ B the event consisting of all outcomes in both A and B . Note that A ∩ B = B ∩ A . 2 Definition: Compliment, A c the event consisting of all outcomes in S that are not in A . Venn diagrams: Definition: disjoint events When events A and B have no outcomes in common, denoted A ∩ B = ∅ , we say A and B are disjoint events. Definition: null event denoted by ∅ , an event with zero outcomes. It cannot occur. Note: All outcomes in the sample space S are disjoint events....
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This note was uploaded on 05/06/2011 for the course STAT 5021 taught by Professor Staff during the Spring '08 term at Minnesota.

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Probability-solutions - Introduction to probability Note...

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