Author: Andrew Dados
Date: 01:18:42 02/01/01
After reading all those posts about statistical significance I decided to run
some tests.
My model is based on win expectancy formula:
D:=1/(1+exp(delta/400)) where delta is rating difference.
For a single match I started with calculating probabilities of 3 events: win,
draw and loss (p1,pd and p0).
Note you have freedom to set draw probability within those limits:
D = p1 + 0.5*pd and
1-D = p0 + 0.5*pd;
I started with initial probability of draw set for 'equal opponents' (delta=0)
then scaled it down by some factor depending on delta.
Then all you need is enumerate through all posible scores and all games....
For equal opponents (delta=0) and match of 5 games you can get (score of 1/2
scaled to 1):
p1/p0/pd = 0.250 0.250 0.500 (50% draws):
score probability
0-->0.098
1-->0.977
2-->4.395
3-->11.719
4-->20.508
5-->24.609
6-->20.508
7-->11.719 <- means: in 5 game match you'll get 3.5 pts with 11.7% chance
8-->4.395
9-->0.977
10-->0.098
p1/p0/pd = 0.350 0.350 0.300 (30% draws):
0-->0.525
1-->2.251
2-->6.485
3-->12.311
4-->18.246
5-->20.364
6-->18.246
7-->12.311 <- less draws; bigger chance of getting 3.5 pts...
8-->6.485
9-->2.251
10-->0.525
p1/p0/pd = 0.500 0.500 0.000 (no draws model):
0-->3.125
1-->0.000
2-->15.625
3-->0.000
4-->31.250
5-->0.000
6-->31.250
7-->0.000
8-->15.625
9-->0.000
10-->3.125
Note big impact of draw chance on resulting model.
Now big tables...
****
Explanation: in below table entry marked < > means:
"For match of 30 games against opponent rated 40 elo better then me
I have 22.20% chances to get 50% score or better."
****
=============================================
1) Expectancy of getting 50% score or better:
==============================================
(Draw probability for equal opponents: 50%)
-----------------------------------------------------------------------------
200 160 120 80 40 0 -40 -80 -120 <-delta
games: ----------------------------------------------------------------------
5: 98.38 96.30 92.29 85.49 75.38 62.30 47.66 33.43 21.42 12.54
10: 99.72 98.86 96.19 89.65 77.14 58.81 38.29 20.77 9.28 3.41
15: 99.95 99.64 98.09 92.63 79.31 57.22 32.53 13.93 4.41 1.03
20: 99.99 99.88 99.03 94.71 81.32 56.27 28.30 9.67 2.18 0.33
25: 100.00 99.96 99.50 96.17 83.11 55.61 24.96 6.85 1.11 0.11
30: 100.00 99.99 99.74 97.21 84.70 55.13 <22.20> 4.91 0.57 0.03
35: 100.00 100.00 99.86 97.96 86.11 54.75 19.87 3.55 0.30 0.01
40: 100.00 100.00 99.93 98.50 87.36 54.45 17.88 2.59 0.16 0.00
45: 100.00 100.00 99.96 98.89 88.48 54.19 16.14 1.90 0.08 0.00
50: 100.00 100.00 99.98 99.18 89.49 53.98 14.62 1.40 0.04 0.00
probability of: -------------------------------------------------------------
win : 0.58 0.51 0.44 0.38 0.31 0.25 0.20 0.15 0.11 0.08
draw : 0.37 0.41 0.44 0.47 0.49 0.50 0.49 0.47 0.44 0.41
loss : 0.06 0.08 0.11 0.15 0.20 0.25 0.31 0.38 0.44 0.51
(Draw probability for equal opponents: 30%)
-----------------------------------------------------------------------------
200 160 120 80 40 0 -40 -80 -120 <-delta
games: ----------------------------------------------------------------------
5: 96.25 93.20 88.33 81.19 71.66 60.18 47.74 35.63 24.99 16.51
10: 98.97 97.19 93.21 85.60 73.44 57.39 40.00 24.54 13.20 6.25
15: 99.70 98.79 95.94 88.90 75.47 56.08 35.06 18.00 7.53 2.58
20: 99.91 99.47 97.54 91.37 77.37 55.28 31.37 13.61 4.46 1.11
25: 99.97 99.76 98.49 93.24 79.08 54.73 28.38 10.46 2.69 0.49
30: 99.99 99.89 99.06 94.67 80.63 54.32 25.88 8.14 1.65 0.22
35: 100.00 99.95 99.41 95.77 82.03 54.01 23.73 6.38 1.02 0.10
40: 100.00 99.98 99.63 96.64 83.30 53.75 21.84 5.04 0.64 0.05
45: 100.00 99.99 99.77 97.31 84.45 53.54 20.17 3.99 0.40 0.02
50: 100.00 100.00 99.85 97.85 85.50 53.36 18.68 3.18 0.25 0.01
probability of: -------------------------------------------------------------
win : 0.65 0.59 0.53 0.47 0.41 0.35 0.29 0.24 0.20 0.16
draw : 0.22 0.24 0.27 0.28 0.30 0.30 0.30 0.28 0.27 0.24
loss : 0.13 0.16 0.20 0.24 0.29 0.35 0.41 0.47 0.53 0.59
(Draw probability for equal opponents: 10%)
-----------------------------------------------------------------------------
200 160 120 80 40 0 -40 -80 -120 <-delta
games: ----------------------------------------------------------------------
5: 93.42 89.43 83.76 76.23 66.94 56.35 45.27 34.59 25.13 17.37
10: 98.10 95.63 90.91 83.00 71.49 57.12 41.75 27.68 16.60 9.03
15: 99.25 97.67 93.78 85.84 72.69 55.18 36.57 20.81 10.10 4.19
20: 99.72 98.82 95.90 88.59 74.65 54.69 33.46 16.65 6.67 2.16
25: 99.89 99.37 97.22 90.63 76.22 54.15 30.69 13.37 4.42 1.11
30: 99.96 99.67 98.11 92.28 77.71 53.81 28.41 10.89 2.98 0.58
35: 99.98 99.82 98.70 93.61 79.05 53.53 26.40 8.93 2.03 0.31
40: 99.99 99.90 99.11 94.69 80.29 53.31 24.64 7.37 1.39 0.17
45: 100.00 99.95 99.38 95.57 81.42 53.12 23.06 6.10 0.96 0.09
50: 100.00 99.97 99.57 96.30 82.47 52.96 21.63 5.08 0.66 0.05
probability of: -------------------------------------------------------------
win : 0.72 0.67 0.62 0.57 0.51 0.45 0.39 0.34 0.29 0.24
draw : 0.07 0.08 0.09 0.09 0.10 0.10 0.10 0.09 0.09 0.08
loss : 0.20 0.24 0.29 0.34 0.39 0.45 0.51 0.57 0.62 0.67
=============================================
2) Expectancy of getting 60% score or better:
==============================================
(Draw probability for equal opponents: 50%)
-----------------------------------------------------------------------------
200 160 120 80 40 0 -40 -80 -120 <-delta
games: ----------------------------------------------------------------------
5: 93.27 87.46 78.58 66.57 52.34 37.70 24.62 14.51 7.71 3.70
10: 96.78 91.41 80.80 64.22 44.04 25.17 11.71 4.38 1.31 0.32
15: 98.44 94.15 83.24 63.57 38.99 18.08 6.09 1.46 0.25 0.03
20: 99.23 95.98 85.39 63.43 35.26 13.41 3.30 0.51 0.05 0.00
25: 99.62 97.21 87.24 63.50 32.26 10.13 1.83 0.18 0.01 0.00
30: 99.81 98.06 88.82 63.68 29.75 7.75 1.03 0.07 0.00 0.00
35: 99.90 98.64 90.18 63.91 27.58 5.98 0.59 0.02 0.00 0.00
40: 99.95 99.04 91.36 64.18 25.68 4.65 0.34 0.01 0.00 0.00
45: 99.97 99.32 92.38 64.47 23.98 3.63 0.19 0.00 0.00 0.00
50: 99.99 99.52 93.27 64.76 22.46 2.84 0.11 0.00 0.00 0.00
probability of: -------------------------------------------------------------
win : 0.58 0.51 0.44 0.38 0.31 0.25 0.20 0.15 0.11 0.08
draw : 0.37 0.41 0.44 0.47 0.49 0.50 0.49 0.47 0.44 0.41
loss : 0.06 0.08 0.11 0.15 0.20 0.25 0.31 0.38 0.44 0.51
(Draw probability for equal opponents: 30%)
-----------------------------------------------------------------------------
200 160 120 80 40 0 -40 -80 -120 <-delta
games: ----------------------------------------------------------------------
5: 89.67 83.49 75.01 64.37 52.26 39.82 28.34 18.81 11.67 6.80
10: 94.02 87.60 77.03 62.30 45.18 28.77 15.88 7.56 3.12 1.12
15: 96.51 90.72 79.33 61.72 40.85 22.18 9.65 3.34 0.92 0.21
20: 97.93 93.00 81.41 61.58 37.62 17.58 6.08 1.54 0.29 0.04
25: 98.75 94.68 83.25 61.63 35.01 14.16 3.91 0.72 0.09 0.01
30: 99.24 95.93 84.86 61.77 32.79 11.54 2.55 0.35 0.03 0.00
35: 99.54 96.87 86.29 61.97 30.87 9.47 1.68 0.17 0.01 0.00
40: 99.72 97.58 87.56 62.19 29.15 7.82 1.11 0.08 0.00 0.00
45: 99.83 98.13 88.70 62.42 27.61 6.49 0.74 0.04 0.00 0.00
50: 99.89 98.54 89.71 62.67 26.21 5.40 0.50 0.02 0.00 0.00
probability of: -------------------------------------------------------------
win : 0.65 0.59 0.53 0.47 0.41 0.35 0.29 0.24 0.20 0.16
draw : 0.22 0.24 0.27 0.28 0.30 0.30 0.30 0.28 0.27 0.24
loss : 0.13 0.16 0.20 0.24 0.29 0.35 0.41 0.47 0.53 0.59
(Draw probability for equal opponents: 10%)
-----------------------------------------------------------------------------
200 160 120 80 40 0 -40 -80 -120 <-delta
games: ----------------------------------------------------------------------
5: 88.55 82.63 74.87 65.41 54.73 43.65 33.06 23.77 16.24 10.57
10: 92.00 85.19 75.00 61.65 46.50 31.73 19.41 10.60 5.18 2.27
15: 94.69 88.08 76.73 60.65 42.19 25.18 12.68 5.35 1.90 0.57
20: 96.49 90.47 78.59 60.39 39.18 20.67 8.65 2.84 0.74 0.15
25: 97.67 92.36 80.33 60.39 36.80 17.27 6.04 1.55 0.29 0.04
30: 98.44 93.85 81.91 60.50 34.79 14.58 4.27 0.86 0.12 0.01
35: 98.95 95.03 83.33 60.66 33.04 12.40 3.05 0.48 0.05 0.00
40: 99.29 95.97 84.62 60.85 31.48 10.60 2.19 0.27 0.02 0.00
45: 99.52 96.72 85.78 61.06 30.07 9.11 1.59 0.15 0.01 0.00
50: 99.67 97.32 86.83 61.28 28.78 7.85 1.15 0.09 0.00 0.00
probability of: -------------------------------------------------------------
win : 0.72 0.67 0.62 0.57 0.51 0.45 0.39 0.34 0.29 0.24
draw : 0.07 0.08 0.09 0.09 0.10 0.10 0.10 0.09 0.09 0.08
loss : 0.20 0.24 0.29 0.34 0.39 0.45 0.51 0.57 0.62 0.67
=============================================
3) Expectancy of getting 75% score or better:
==============================================
(Draw probability for equal opponents: 50%)
-----------------------------------------------------------------------------
200 160 120 80 40 0 -40 -80 -120 <-delta
games: ----------------------------------------------------------------------
5: 79.82 68.98 55.78 41.57 28.18 17.19 9.38 4.56 1.98 0.78
10: 65.63 47.61 29.55 15.19 6.30 2.07 0.53 0.11 0.02 0.00
15: 71.82 50.50 28.39 12.08 3.72 0.81 0.12 0.01 0.00 0.00
20: 63.96 38.67 16.97 5.03 0.95 0.11 0.01 0.00 0.00 0.00
25: 69.62 41.72 16.95 4.23 0.60 0.05 0.00 0.00 0.00 0.00
30: 63.71 33.13 10.55 1.84 0.16 0.01 0.00 0.00 0.00 0.00
35: 68.76 35.85 10.69 1.58 0.10 0.00 0.00 0.00 0.00 0.00
40: 63.86 29.04 6.80 0.70 0.03 0.00 0.00 0.00 0.00 0.00
45: 68.43 31.43 6.94 0.61 0.02 0.00 0.00 0.00 0.00 0.00
50: 64.16 25.79 4.48 0.27 0.01 0.00 0.00 0.00 0.00 0.00
probability of: -------------------------------------------------------------
win : 0.58 0.51 0.44 0.38 0.31 0.25 0.20 0.15 0.11 0.08
draw : 0.37 0.41 0.44 0.47 0.49 0.50 0.49 0.47 0.44 0.41
loss : 0.06 0.08 0.11 0.15 0.20 0.25 0.31 0.38 0.44 0.51
(Draw probability for equal opponents: 30%)
-----------------------------------------------------------------------------
200 160 120 80 40 0 -40 -80 -120 <-delta
games: ----------------------------------------------------------------------
5: 76.26 66.67 55.44 43.41 31.75 21.57 13.58 7.92 4.30 2.19
10: 63.98 48.72 33.11 19.67 10.04 4.36 1.61 0.51 0.14 0.03
15: 69.19 50.99 31.94 16.40 6.71 2.15 0.54 0.10 0.02 0.00
20: 62.39 40.91 21.30 8.39 2.41 0.50 0.07 0.01 0.00 0.00
25: 67.20 43.45 21.24 7.35 1.71 0.26 0.03 0.00 0.00 0.00
30: 62.09 36.03 14.72 3.92 0.64 0.06 0.00 0.00 0.00 0.00
35: 66.39 38.35 14.84 3.49 0.47 0.03 0.00 0.00 0.00 0.00
40: 62.16 32.38 10.50 1.90 0.18 0.01 0.00 0.00 0.00 0.00
45: 66.06 34.46 10.65 1.72 0.13 0.00 0.00 0.00 0.00 0.00
50: 62.38 29.43 7.64 0.95 0.05 0.00 0.00 0.00 0.00 0.00
probability of: -------------------------------------------------------------
win : 0.65 0.59 0.53 0.47 0.41 0.35 0.29 0.24 0.20 0.16
draw : 0.22 0.24 0.27 0.28 0.30 0.30 0.30 0.28 0.27 0.24
loss : 0.13 0.16 0.20 0.24 0.29 0.35 0.41 0.47 0.53 0.59
(Draw probability for equal opponents: 10%)
-----------------------------------------------------------------------------
200 160 120 80 40 0 -40 -80 -120 <-delta
games: ----------------------------------------------------------------------
5: 71.13 62.13 52.08 41.59 31.46 22.44 15.05 9.48 5.63 3.15
10: 61.58 48.14 34.44 22.20 12.74 6.45 2.87 1.12 0.39 0.12
15: 67.70 51.49 34.49 19.83 9.58 3.84 1.27 0.34 0.08 0.01
20: 61.38 42.37 24.47 11.42 4.19 1.19 0.26 0.04 0.01 0.00
25: 65.44 44.40 24.25 10.17 3.16 0.71 0.11 0.01 0.00 0.00
30: 60.89 37.83 17.94 6.13 1.45 0.23 0.02 0.00 0.00 0.00
35: 64.75 39.90 18.05 5.59 1.12 0.14 0.01 0.00 0.00 0.00
40: 60.96 34.54 13.60 3.44 0.53 0.05 0.00 0.00 0.00 0.00
45: 64.43 36.39 13.74 3.16 0.41 0.03 0.00 0.00 0.00 0.00
50: 61.13 31.84 10.48 1.97 0.20 0.01 0.00 0.00 0.00 0.00
probability of: -------------------------------------------------------------
win : 0.72 0.67 0.62 0.57 0.51 0.45 0.39 0.34 0.29 0.24
draw : 0.07 0.08 0.09 0.09 0.10 0.10 0.10 0.09 0.09 0.08
loss : 0.20 0.24 0.29 0.34 0.39 0.45 0.51 0.57 0.62 0.67
==============================================
3) Expectancy of getting 100% score:
==============================================
(Draw probability for equal opponents: 50%)
-----------------------------------------------------------------------------
200 160 120 80 40 0 -40 -80 -120 <-delta
games: ----------------------------------------------------------------------
5: 6.41 3.50 1.72 0.75 0.29 0.10 0.03 0.01 0.00 0.00
10: 0.41 0.12 0.03 0.01 0.00 0.00 0.00 0.00 0.00 0.00
15: 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
20: 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
25: 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
30: 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
35: 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
40: 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
45: 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
50: 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
probability of: -------------------------------------------------------------
win : 0.58 0.51 0.44 0.38 0.31 0.25 0.20 0.15 0.11 0.08
draw : 0.37 0.41 0.44 0.47 0.49 0.50 0.49 0.47 0.44 0.41
loss : 0.06 0.08 0.11 0.15 0.20 0.25 0.31 0.38 0.44 0.51
(Draw probability for equal opponents: 30%)
-----------------------------------------------------------------------------
200 160 120 80 40 0 -40 -80 -120 <-delta
games: ----------------------------------------------------------------------
5: 11.62 7.34 4.29 2.31 1.15 0.53 0.22 0.09 0.03 0.01
10: #1.35# 0.54 0.18 0.05 0.01 0.00 0.00 0.00 0.00 0.00
15: 0.16 0.04 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00
20: 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
25: 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
30: 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
35: 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
40: 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
45: 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
50: 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
probability of: -------------------------------------------------------------
win : 0.65 0.59 0.53 0.47 0.41 0.35 0.29 0.24 0.20 0.16
draw : 0.22 0.24 0.27 0.28 0.30 0.30 0.30 0.28 0.27 0.24
loss : 0.13 0.16 0.20 0.24 0.29 0.35 0.41 0.47 0.53 0.59
# # entry: if I am 200 Elo better I still have only 1.35% chance of beating my
opp 10-0
==========================
-Andrew-
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