Probability

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

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= x=100 x=60 Neither=10

(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= 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

(c) Given that a player plays softball, what is the probability that he plays both soccer and softball Total=100 Soccer=80 Softball= 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

(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

(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

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

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

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)

Counting problem using Excel

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