Cumulative frequency distribution of past record to calculate estimated P

[Joe's]

Hi!

PS. I have put the problem in a google sheets document in a more readable format, that you can freely edit, and play with. https://docs.google.com/spreadsheets/d/1NGxP1hHFcYXPh8Sy2w_52FSFAURIDRo83utXp97uHp0/edit?usp=sharing

This is the problem I am stuck with:

Runner : Speed cumulative frequency best ---> worst

Percentile (top x percentile)
1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99
runner
A 0.00 0.00 0.38 0.50 0.50 0.50 0.63 0.63 0.63 0.63 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.88 0.88 0.88
B 0.00 0.00 0.25 0.25 0.42 0.50 0.75 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 1.00
C 0.00 0.00 0.00 0.00 0.23 0.31 0.46 0.46 0.46 0.46 0.54 0.62 0.62 0.69 0.69 0.77 0.85 0.92 0.92 1.00 1.00
D 0.00 0.00 0.13 0.13 0.13 0.25 0.38 0.63 0.63 0.88 0.88 0.88 0.88 0.88 1.00 1.00 1.00 1.00 1.00 1.00 1.00
E 0.00 0.00 0.00 0.00 0.00 0.14 0.14 0.29 0.43 0.43 0.43 0.43 0.43 0.57 0.71 0.71 0.86 0.86 1.00 1.00 1.00
F 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.07 0.14 0.14 0.14 0.21 0.21 0.36 0.43 0.43 0.43 0.64 0.79 0.86 0.93
G 0.00 0.00 0.00 0.00 0.25 0.25 0.25 0.25 0.25 0.38 0.63 0.63 0.63 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.88
H 0.00 0.00 0.00 0.00 0.00 0.20 0.60 0.60 0.60 0.80 0.80 0.80 0.80 0.80 1.00 1.00 1.00 1.00 1.00 1.00 1.00

(numbers show frequency of runs within that percentile, eg 0.38 means 38% of the runs that runner made were within that percentile)

From here, how does one calculate most accurately, what the probability each runner has of coming first, second, third, fourth, fifth, sixth, seventh, eighth?

	1st	2nd	3rd	4th	5th	6th	7th	8th
A
B
C
D
E
F
G
H

Thanks everyone, have been stuck with this problem for ages now, I will not describe what calculations I have tried so far because it may not be the best way at all.

Best Regards,
Joe

 

[Joe's]

Using frequency of past speeds achieved to calculate P of runners finishing positions

Hi all!

I am stuck with this problem:

8 runners are to run a race, their past records have been examined to determine the frequency in which they have achieved being within the top x percentile.


Top x Percentile

Top 1%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

55%

60%

65%

70%

75%

80%

85%

90%

95%

99%

0.00 0.00 0.38 0.50 0.50 0.50 0.63 0.63 0.63 0.63 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.88 0.88 0.88 A
0.00 0.00 0.25 0.25 0.42 0.50 0.75 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 1.00 B
0.00 0.00 0.00 0.00 0.23 0.31 0.46 0.46 0.46 0.46 0.54 0.62 0.62 0.69 0.69 0.77 0.85 0.92 0.92 1.00 1.00 C
0.00 0.00 0.13 0.13 0.13 0.25 0.38 0.63 0.63 0.88 0.88 0.88 0.88 0.88 1.00 1.00 1.00 1.00 1.00 1.00 1.00 D Runners
0.00 0.00 0.00 0.00 0.00 0.14 0.14 0.29 0.43 0.43 0.43 0.43 0.43 0.57 0.71 0.71 0.86 0.86 1.00 1.00 1.00 E
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.07 0.14 0.14 0.14 0.21 0.21 0.36 0.43 0.43 0.43 0.64 0.79 0.86 0.93 F
0.00 0.00 0.00 0.00 0.25 0.25 0.25 0.25 0.25 0.38 0.63 0.63 0.63 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.88 G
0.00 0.00 0.00 0.00 0.00 0.20 0.60 0.60 0.60 0.80 0.80 0.80 0.80 0.80 1.00 1.00 1.00 1.00 1.00 1.00 1.00 H



From here, calculate in the most accurate way possible, the probability of each runner finishing 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th

Thank you for your help!


Best Regards,
Joe

 

[Deleted member 4993]

Hi!

This is the problem I am stuck with:

Runner : Speed cumulative frequency best ---> worst

A 0.00 0.00 0.38 0.50 0.50 0.50 0.63 0.63 0.63 0.63 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.88 0.88 0.88B 0.00 0.00 0.25 0.25 0.42 0.50 0.75 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 1.00
C 0.00 0.00 0.00 0.00 0.23 0.31 0.46 0.46 0.46 0.46 0.54 0.62 0.62 0.69 0.69 0.77 0.85 0.92 0.92 1.00 1.00
D 0.00 0.00 0.13 0.13 0.13 0.25 0.38 0.63 0.63 0.88 0.88 0.88 0.88 0.88 1.00 1.00 1.00 1.00 1.00 1.00 1.00
E 0.00 0.00 0.00 0.00 0.00 0.14 0.14 0.29 0.43 0.43 0.43 0.43 0.43 0.57 0.71 0.71 0.86 0.86 1.00 1.00 1.00
F 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.07 0.14 0.14 0.14 0.21 0.21 0.36 0.43 0.43 0.43 0.64 0.79 0.86 0.93
G 0.00 0.00 0.00 0.00 0.25 0.25 0.25 0.25 0.25 0.38 0.63 0.63 0.63 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.88
H 0.00 0.00 0.00 0.00 0.00 0.20 0.60 0.60 0.60 0.80 0.80 0.80 0.80 0.80 1.00 1.00 1.00 1.00 1.00 1.00 1.00

From here, how does one calculate most accurately, what the probability each runner has of coming first, second, third, fourth, fifth, sixth, seventh, eighth?

	1st	2nd	3rd	4th	5th	6th	7th	8th
A
B
C
D
E
F
G
H

Thanks everyone, have been stuck with this problem for ages now, I will not describe what calculations I have tried so far because it may not be the best way at all.

Best Regards,
Joe

What are your thoughts?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/for

 

[Joe's]

What are your thoughts?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/for

Thank you for the reply Subhotosh!

Yes, I am stuck with definitions, and unsure how to declare the problem in the standard way.

Here is the method I have found so far, although it is inaccurate, there is an error, or something I am not accounting for.

Where the sum of the frequencies for all runners reach 1 or just above under a certain percentile, the frequency is taken for each runner, and divided by the sum of frequencies for all runners runs being within that percentile. This should give the probability of achieving first place for each runner.

In this case it is under the 5th column (within top 20 percentile) that frequencies add up to 1 or above.

frequency of runs within top 20 percentile for:
A = 0.5
B = 0.42
C = 0.23
D = 0.13
E = 0
F = 0
G = 0.25

Sum of frequencies of runs being within top 20 percentile = 0.5 + 0.42 + 0.23 + 0.13 + 0 + 0 + 0.25 = 1.53
Therefore Pprobabilty (P) of runner (A..G) finishing first (1st):
P(A1st) = 0.5 / 1.53 = 0.33
P(B1st) = 0.42 / 1.53 etc...

Then the problem becomes further complicated for second place... for each runner does one then multiply the frequency of it's runs at the percentile where the frequency of all runners' runs add up to 2 (or just above), by P(runner2nd) * (1 - P(runner1st)) ?

Also to be noted, is that using this method, the P for runner finishing 1st..8th does not add up to 1, nor do the P for all runners finishing in each position add up to 1.