a. Show the probability of teams ranking 2 and 15 playing game 1 and Teams 7 and 10. Plot the PDF for each in pairs (2/15, 7/10).

Solution:

Let the binomial probability is b(n, x) which is given by

n is total number of trial p is probability of success(scoring) which is given by P(k)=0.5*(1.02^-(k^1.5)) where k is ranking of the teams q is the probability of failure which is given by q=1-p x is successful trial which is given by n*p b is binomial probability

Point is total score of the team in a game which is given by 2*x.

The table shown below shows the above parameters for all teams.

Team x Point n P q b

1

37

74

75

0.490

0.510

0.0917

2

36

72

75

0.473

0.527

0.0912

3

34

68

75

0.451

0.549

0.0922

4

33

66

75

0.427

0.573

0.0900

5

31

62

75

0.401

0.599

0.0909

6

29

58

75

0.374

0.626

0.0918

7

26

52

75

0.346

0.654

0.0964

8

24

48

75

0.319

0.681

0.0984

9

22

44

75

0.293

0.707

0.1007

10

21

42

75

0.267

0.733

0.0991

11

19

38

75

0.243

0.757

0.1031

12

17

34

75

0.220

0.780

0.1082

13

15

30

75

0.198

0.802

0.1143

14

14

28

75

0.177

0.823

0.1149

15

12

24

75

0.158

0.842

0.1247

16

11

22

75

0.141

0.859

0.1278

Total score of the first game is shown below

Teams’ rank

Team(i)

Team(j)

Point(i)

Point(j)

1

16

74

22

2

15

71

24

3

14

68

27

4

13

65

30

5

12

61

33

6

11

57

37

7

10

52

41

8

9

48

44

Matlab code to plot PDF for all games in pair is shown below

clear all; clc;

N=75;

for i=1:8 figure for k=i:16-(2*i-1):17-i p=.5*(1.02^-(k^1.5)); %Probability based on ranking S=round(N*p); %successful scoring pt=2*S; %point for i=1:75

Ki=round(i*p); %

B(i)=NchooseK(N,i)*p^(i)*(1-p)^(N-i);

end plot(B) hold on xlabel('successful shoot') ylabel('binomial Probability') title('Pair teams') end end

Output of the code

Plots of PDF for each in pair, 1/16, 2/15, 3/14, 4/13, 5/12, 6/11, 7/10, and 8/9 are shown below respectively.

Fig: Team 1 and 16 Fig: Team 2 and 15

Fig: Team 3 and 14 Fig: Team 4 and 13

Fig: Team 5 and 12 Fig: Team 6 and 11