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Probability

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Discussion Topics Mutually exclusive events Addition rule Conditional probability Independent event Counting problem

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Question 1 A high school has 100 students. 80 of them play soccer.70 play softball. 10 play neither of the two. (a) How many students play both soccer and softball. Total=100 Soccer=80 Softball=70 80-xx70-x 80-x+x+70-x+10=100 160-x=100 x=60 Neither=10

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(b) A student is chosen randomly, what is the probability that the student plays only soccer or only softball? Is the event mutually exclusive? Total=100 Soccer=80 Softball=70 80-606070-60 Neither=10 P(A)=The event of a student playing only soccer P(B)=The event of a student playing only softball P(C)=The event of a student playing only soccer or only softball P(C)=P(A)+P(B) P(C)=20/100+10/100 P(C)=0.3 The events are mutually exclusive because P(A∩B) = 0

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(c) Given that a player plays softball, what is the probability that he plays both soccer and softball Total=100 Soccer=80 Softball=70 80-606070 Neither=10 P(A)=The event of a student playing softball P(B)=The event of a student playing both softball and Soccer P(B|A)= 60/70=0.857

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(d) Determine whether these events are independent: (i) The event of playing only soccer (ii) The event of playing only softball Total=100 Soccer=80 Softball=70 Only soccer 20 Only softball 10 Neither=10 P(A)=The event of a student playing only soccer P(B)=The event of a student playing only softball For independent events: P(A|B) = P(A) ; P(A)*P(B)= P(A∩B) 0 ≠ 20/100 ; 20/100*10/100 ≠ 0 Therefore the events are not independent 60

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(d) Determine whether these events are independent: (i) The event of a student playing only softball (ii) The event of a student playing neither sport Total=100 Soccer=80 Softball=70 Only soccer 20 Only softball 10 Neither=10 P(A)=The event of a student playing only softball P(B)=The event of a student playing neither sport For independent events: P(A|B) = P(A) ; P(A)*P(B)= P(A∩B) 0 ≠10/100 ;10/100*10/100 ≠ 0 The events are not independent 60

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Counting problem You have just been hired as a book representative for Pearson Education. On your first day, you must travel to seven schools to introduce yourself. How many different routes are possible? Solution: We represent the schools by A,B,C,D,E,F and G : School A be visited 1 st, 2 nd, 3 rd, 4 th, 5 th, 6 th, or 7 th : 7 choices School B can be visited 2 nd,3 rd,4 th,5 th,6 th,7 th : 6 choices School C can be visited 3 rd,4 th,5 th,6 th,7 th : 5 choices School D can be visited 4 th,5 th,6 th,7 th :4 choices School E can be visited 5 th,6 th,7 th : 3 choices School F can be visited 6 th,7 th :2 choices School G can be visited 7 th : 1 choice The total number of different routes possible are: 7*6*5*4*3*2*1 = 7! = 5,040

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Counting problem using Excel Solution: We represent the schools by A,B,C,D,E,F and G : We enter the schools together with the number of choices of routes

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Counting problem using Excel Calculate the different possible routes by calculating the factorial using the FACT function. First select cell for displaying fact function Go to the formula bar, type “=” followed by “FACT” FACT(7)

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Counting problem using Excel

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CSTEM Web link http://www.cis.famu.edu/~cdellor/math/ Presentation slides Problems Guidelines Learning materials

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