Project 2 Essay

Submitted By eddy_098
Words: 797
Pages: 4

4. (Extra) Assume that any game consist of N events (shots) per team. Assume N=75 and assume only 2 point shots and no fouls. Assume a team has a shooting percentage related to its Ranking as P(K)= .5*(1.02^-(K^1..5)) with ranking K. Assume that the Scoring Probability Pb(N,P(K)) with follow a binomial distribution with a Expected Value (score) Sc=N*P(K).
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…