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Unformatted text preview: J. Chen, Handout 1, STAT550, 2010 1 STAT550 Applied Probability http://wwwrohan.sdsu.edu/ jchenyp/STAT5502010.htm Chapter 1. Basic Probability 1.1. Sample space and events 1) Experiment and outcomes An EXPERIMENT is a repeatable process that generates outcomes. Each time the experiment is run, one outcome, w , is obtained. The number of outcomes in an experiment may be finite or infinite. 2) Sample space and event The set of all possible outcomes for an experiment, S , is call the SAMPLE SPACE. We also call the outcome in sample space as the sample point. If the outcomes are finite, then S = { w 1 ,w 2 , ,w n } . An EVENT is any subset of the sample space S . Events are usually designated by capital letters ( A,B,C, ). Notice that the entire sample space S and the null set are events. Furthermore, there are the following two relationship between events A B w A w B A = B A B and B A EX1: Let the sample space S defined as: S = { w 1 ,w 2 ,w 3 } . Find all possible events in S . Obviously, all events defined in S are: ;( w 1 ) , ( w 2 ) , ( w 3 );( w 1 ,w 2 ) , ( w 1 ,w 3 ) , ( w 2 ,w 3 ); and ( w 1 ,w 2 ,w 3 ) Ex2: (Roll a die) An experiment consists of rolling a balanced die one time. What are the outcomes ? and what is the sample space? Solution: Outcomes: 1 , 2 , 3 , 4 , 5 , 6; Sample space S = { 1 , 2 , 3 , 4 , 5 , 6 } . Now we Define events as follows: J. Chen, Handout 1, STAT550, 2010 2 A =outcome is even = { 2 , 4 , 6 } B =outcome is prime number = { 2 , 3 , 5 } C =outcome is 3 = { 1 , 2 , 3 } D =outcome is 1 = { 1 } *Ex3: (Roll Two Dice) Imagine rolling two dice, the first one red, the second one green. Solution: Since each sample outcome is an ordered pair, and the entire sample space can be represented as a 6 6 matrix. Therefore, the sample ample space S = { ( i,j ) ,i,j = 1 , 2 , 3 , 4 , 5 , 6 } ; and events can be defined as follows: A = sum of the faces showing is a 7 = { (1 , 6) , (2 , 5) , (3 , 4) , (4 , 3) , (5 , 2) , (6 , 1) } B = the event that the two faces themselves are odd = { (1 , 1) , (1 , 3) , (1 , 5) , (3 , 1) , (3 , 3) , (3 , 5) , (5 , 1) , (5 , 3) , (5 , 5) } Ex4: (Flip a coin) Consider the experiment of flipping a coin three times. Solution: Since each sample outcome is an ordered triple, its components repre senting the outcomes of the first, second, and third tosses, respectively. So Sample space S = { HHH,HHT,HTH,THH,HTT,THT,TTH,TTT } , and De fine events A = Majority of coins show heads = { HHH,HHTHTH,THH } B = Majority of coins show tails = { TTT,TTH,THT,HTT } 3). Operations on Events Def. 1: Let A and B be any two events defined over the sample space S . Then a. The INTERSECTION of A and B is the event whose outcomes belong to both A and B , denoted by A B . See the shade region on Figure 1 (a), A B = { w : w A and w B } J. Chen, Handout 1, STAT550, 2010 3 b. The UNION of A and...
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