From one point of view, this seems a silly question. There are exactly $4^n$ possible strings of length $n$. Each string occurs equally infrequently: 1 in $4^n$ times. However, if I look at another level, I would expect to have about equal distribution of each letter. For this reason, I will use the word "structuredness" instead of the word "random" when referring to the property I am investigating.
Let $\sigma$ represent any of the $a^k$ strings of length $k$. Let $\sigma(S)$ equal the number of times $\sigma$ occurs as a substring of $S$. Define $d_k = d_k (S)$ as $$d_k=\sum_{\sigma = \textbf{AA}\cdots\textbf{A}}^{\textbf{DD}\cdots\textbf{D}} \sigma(S)^2,$$
where the sum is over all strings of length $k$. To incorporate the a measure of $S$'s structuredness at every level, we define $$d=\sum_{k=1}^{n} d_k.$$
$d_1(S)=2^2+2^2+2^2+2^2=16$ $d_2(S)=2^2+2^2+2^2+1^2=13$ $d_3(S)=2^2+2^2+1^2+1^2=10$ $d_4(S)=2^2+1^2+1^2+1^2=7$ $d_5(S)=1^2+1^2+1^2+1^2=4$ $d_6(S)=1^2+1^2+1^2=3$ $d_7(S)=1^2+1^2=2$ $d_8(S)=1^2=1$ | $d_1(T)=2^2+2^2+2^2+2^2=16$ $d_2(T)=2^2+1^2+1^2+1^2+1^2+1^2=9$ $d_3(T)=1^2+1^2+1^2+1^2+1^2+1^2=6$ $d_4(T)=1^2+1^2+1^2+1^2+1^2=5$ $d_5(T)=1^2+1^2+1^2+1^2=4$ $d_6(T)=1^2+1^2+1^2=3$ $d_7(T)=1^2+1^2=2$ $d_8(T)=1^2=1$ |
Note that this definition of $d$ makes no reference to the alphabet from which the letters are selected. The same calculations would apply if our alphabet contained one hundred letters. This flexibility greatly aids in calculation.
I selected only win-loss records that had an even number of wins & losses, since I wanted to compare them more easily with the home-away records which have an even number of home and away games. Likewise for the binary strings.
- I expect the win-loss records to be close to a 0.5 relative structuredness.
- I expect the home-away schedule to have a high relative structuredness, since I expect that teams will not have more than three home games or three away games in a row..
- If they are truly random, I expect the random strings to have relative structuredness even closer to 0.5 than the win loss records have.
The win-loss records has relative structuredness = 0.406
The home-away schedule has relative structuredness = 0.718
The random strings have relative structuredness = 0.244
The win-loss and home-away relative structuredness agree with my hypothesis, but I was surprised .to see the low score from the random strings.
Miami Dolphins 2014
WLLWLWWWLWLWLLWL
d - value: 334
r.s.: 0.500
SF 49ers 2014
WLLWWWLLWWWLLLLW
d - value: 342
r.s.: 0.598
NY Jets 2013
WLWLWLWLWLLLWLWW
d - value: 400
r.s.: 0.906
Dallas Cowboys 2013
WLWLLWWLWLWWLLWL
d - value: 328
r.s.: 0.400
Pittsburgh Steelers 2012
LWLWLWWWWLLWLLLW
d - value: 308
r.s.: 0.042
NY Jets 2011
WWLLLWWWLLWWWLLL
d - value: 344
r.s.: 0.619
Chicago Bears 2011
WLLWLWWWWWLLLLLW
d - value: 312
r.s.: 0.111
Jacksonville Jaguars 2010
WLLWWLLWWWLWWLLL
d - value: 322
r.s.: 0.285
Tennessee Titans 2009
LLLLLLWWWWWLWWLW
d - value: 332
r.s.: 0.467
Carolina Panthers 2009
LLLWWLWLWLLWLWWW
d - value: 312
r.s.: 0.111
Average r.s.: 0.406
Miami Dolphins 2014
HAHAHAAHAHAAHAHH
d - value: 394
r.s.: 0.893
SF 49ers 2014
AHAHHAAHAAHHAAHH
d - value: 334
r.s.: 0.500
NY Jets 2013
HAHAAHHAHAAHHAHA
d - value: 380
r.s.: 0.845
Dallas Cowboys 2013
HAHAHHAAHAAHAHAH
d - value: 354
r.s.: 0.716
Pittsburgh Steelers 2012
AHAHAAHAHHAAHAHH
d - value: 350
r.s.: 0.683
NY Jets 2011
HHAAAHHAHAHAHAHA
d - value: 372
r.s.: 0.721
Chicago Bears 2011
HAHHAHAAHAHHAHAA
d - value: 356
r.s.: 0.732
Jacksonville Jaguars 2010
HAHHAHAAHHAAHAHA
d - value: 328
r.s.: 0.400
Tennessee Titans 2009
AHAAHAHAHAHAHHHA
d - value: 386
r.s.: 0.867
Carolina Panthers 2009
HAAHAHAAHHAHAHAH
d - value: 356
r.s.: 0.732
Average r.s.: 0.718
String 1
0101000011111001
d - value: 306
r.s.: 0.024
String 2
0100000101011111
d - value: 316
r.s.: 0.173
String 3
1101101100100010
d - value: 322
r.s.: 0.285
String 4
1000001101111100
d - value: 318
r.s.: 0.208
String 5
0111000111010001
d - value: 314
r.s.: 0.143
String 6
1011000111011000
d - value: 328
r.s.: 0.400
String 7
1111100101001000
d - value: 320
r.s.: 0.246
String 8
0101011000101110
d - value: 322
r.s.: 0.285
String 9
1111010000010101
d - value: 316
r.s.: 0.173
String 10
1001100110101001
d - value: 334
r.s.: 0.500
Average r.s.: 0.244