What Fraction of Possible Letter Sequences Are Words?

Get the lengths of words in the word list for English:

In[1]:=

wordlengths=StringLength[WordList[]];Short[wordlengths,3]

Out[1]//Short=

{1,3,8,5,6,5,7,7,9,11,5,9,5,7,9,5,9,8,4,6,5,5,10,11,12,8,10,7,9,6,9,9,8,5,4,8,10,4,7,8,5,10,9,8,5,7,7,6,9,8,10,6,6,7,8,8,6,4,6,8,4,8,10,8,39048,7,4,5,5,3,5,6,4,4,3,6,4,6,8,7,9,5,4,3,6,6,8,4,6,4,7,9,5,4,6,6,5,7,4,9,4,6,6,3,6,5,6,9,3,6,5,6,8,6,5,4,6,3,10,9,7,4,8,6,8,8,6,6,7}

Make a histogram of their lengths:

In[2]:=

Histogram[wordlengths]

Out[2]=

There are only two words of length 1 in the word list:

In[3]:=

Select[WordList[],StringLength[#]1&]

Out[3]=

{a,I}

Here are the words of length 2:

In[4]:=

Select[WordList[],StringLength[#]2&]

Out[4]=

{ad,ah,am,an,as,at,ax,be,by,dB,do,eh,em,en,er,ex,fa,go,ha,he,hi,hm,ho,id,if,in,IQ,is,it,kc,kW,la,lo,ma,me,mi,ms,mu,my,no,nu,of,oh,on,ow,ox,pa,pH,pi,re,sh,so,ta,ti,to,TV,um,up,us,we,xi,ya,ye}

(Some of them are slightly questionable words…)

Here is a count of the number of length-2 words in the word list:

In[5]:=

Length[Select[WordList[],StringLength[#]2&]]

Out[5]=

Here is a list of all possible pairs of letters:

In[6]:=

StringJoin/@Tuples[Alphabet[],2]//Short

Out[6]//Short=

{aa,ab,ac,ad,ae,af,ag,ah,ai,aj,ak,654,zp,zq,zr,zs,zt,zu,zv,zw,zx,zy,zz}

Find the number of possible pairs:

In[7]:=

Length[%]

Out[7]=

676

One can compute the number of possible pairs like this:

In[8]:=

26^2

Out[8]=

676

Note that the number of actual 2-letter words is much smaller than the number of combinations of 2 letters.

The list of possible 3-letter combinations:

In[9]:=

StringJoin/@Tuples[Alphabet[],3]

Out[9]=

aaa,aab,aac,aad,aae,aaf,aag,aah,aai,aaj,aak,aal,aam,aan,aao,aap,aaq,aar,aas,aat,aau,aav,aaw,aax,aay,aaz,aba,abb,abc,abd,abe,abf,abg,abh,abi,abj,abk,abl,abm,abn,abo,abp,abq,abr,abs,abt,abu,abv,abw,abx,aby,abz,aca,acb,acc,acd,ace,acf,acg,ach,

⋯17456⋯

,zxs,zxt,zxu,zxv,zxw,zxx,zxy,zxz,zya,zyb,zyc,zyd,zye,zyf,zyg,zyh,zyi,zyj,zyk,zyl,zym,zyn,zyo,zyp,zyq,zyr,zys,zyt,zyu,zyv,zyw,zyx,zyy,zyz,zza,zzb,zzc,zzd,zze,zzf,zzg,zzh,zzi,zzj,zzk,zzl,zzm,zzn,zzo,zzp,zzq,zzr,zzs,zzt,zzu,zzv,zzw,zzx,zzy,zzz

Full expression not available

(

original memory size:

0.7 MB)

A count of them:

In[10]:=

26^3

Out[10]=

17576

Here is the number of actual 3-letter words in the word list:

In[11]:=

Length[Select[WordList[],StringLength[#]3&]]

Out[11]=

596

The count of possible 4-letter sequences:

In[12]:=

26^4

Out[12]=

456976

The actual number of 4-letter words:

In[13]:=

Length[Select[WordList[],StringLength[#]4&]]

Out[13]=

1914

Compare the amounts of letter sequences and valid words by length in log scale:

In[14]:=

ListLogPlotTranspose[Table[{26^n,Length[Select[WordList[],StringLength[#]n&]]},{n,7}]],



Out[14]=

	Letter Sequences
	Words

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What Fraction of Possible Letter Sequences Are Words?

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What Fraction of Possible Letter Sequences Are Words?

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