In the rally point scoring method
now widely adopted in volleyball,
either team can score a point by
winning the rally after a serve,
regardless of who serves the ball.
The first team to earn 25 points
wins the set, except that a team
must win by at least two points —
that is, if the score is 24-24, then
the final score must be 26-24,
27-25, 28-26, etc.
Assume two
evenly matched teams are playing each other.
To at least four significant figures, what is the expected
number of rallies (total points
scored) per set?
clearvars,clc
scoreCounts=zeros(1,200);
for i=1:10
points=0;
for trial=1:10000000
score=zeros(1,2);
while max(score)<25 || abs(score(1)-score(2))<2
r=randi([1 2]);
score(r)=score(r)+1;
end
points=points+sum(score);
scoreCounts(sum(score))=scoreCounts(sum(score))+1;
end
points/trial
end
f=find(scoreCounts);
mn=min(f);mx=max(f);
accum=0;
for i=mn:mx
accum=accum+scoreCounts(i);
fprintf('%3d %10d %10d\n',i,scoreCounts(i),accum)
end
does ten sets of 10 million trials each. The average score for each set is shown:
ans =
44.7295996
ans =
44.7293298
ans =
44.729314
ans =
44.7297298
ans =
44.731172
ans =
44.7302784
ans =
44.7285294
ans =
44.7285804
ans =
44.7325368
ans =
44.7292464
So the answer to four significant figures is 44.73.
A distribution of total scores is also shown:
score number of cumulative
cases number
25 2 2
26 68 70
27 482 552
28 2147 2699
29 7624 10323
30 22021 32344
31 55658 88002
32 122431 210433
33 244437 454870
34 448100 902970
35 762761 1665731
36 1212694 2878425
37 1822024 4700449
38 2594801 7295250
39 3517733 10812983
40 4572384 15385367
41 5717066 21102433
42 6893292 27995725
43 8039809 36035534
44 9097583 45133117
45 10011436 55144553
46 10731378 65875931
47 11212966 77088897
48 11458228 88547125
49 0 88547125
50 5728355 94275480
51 0 94275480
52 2863590 97139070
53 0 97139070
54 1430352 98569422
55 0 98569422
56 714844 99284266
57 0 99284266
58 357922 99642188
59 0 99642188
60 179304 99821492
61 0 99821492
62 89303 99910795
63 0 99910795
64 44511 99955306
65 0 99955306
66 22277 99977583
67 0 99977583
68 11130 99988713
69 0 99988713
70 5613 99994326
71 0 99994326
72 2833 99997159
73 0 99997159
74 1405 99998564
75 0 99998564
76 732 99999296
77 0 99999296
78 350 99999646
79 0 99999646
80 175 99999821
81 0 99999821
82 98 99999919
83 0 99999919
84 40 99999959
85 0 99999959
86 18 99999977
87 0 99999977
88 12 99999989
89 0 99999989
90 4 99999993
91 0 99999993
92 5 99999998
93 0 99999998
94 2 100000000
|
|
Posted by Charlie
on 2024-12-20 09:27:47 |
computer solution
