
import matplotlib.pyplot as plt
import seaborn as sns


#Probabilty that the character index is a space.
space_p = {1 : 0,
           2 : 0,
           3 : 0,
           4 : 0.5,
           5 : 0.5,
           6 : 0,
           7 : 0,
           11 : 0}



#Make the line very, very, long.
for i in range(4, 501) :
    p = space_p[i]
    
    #If it has a space
    if p != 0 :
        #Half the time 4 characters away will be a space,
        #half the time 5 characters away will be a space.
        for j in (4,5) :
            #Exists already.
            if (i+j) in space_p.keys() :
                space_p[i+j] += 0.5*p
            #New.
            else :
                space_p[i+j] = 0.5*p
                

                
#Expected length.
EL = 0
ELs = {}
#Current probability.
P = 1

#Where does the river stop?
for i in range(12, 501):
    
    #length
    L = i-11
    
    #Next
    N = space_p[i+1]
    
    #EL increases by current length * current probability * probability of letter for this character.
    EL += L*P*(1-N)
    #Dict
    ELs[i] = EL
    #Or this character is a space, update probability.
    P *= N

ELs

{12: 0.625,
 13: 1.09375,
 14: 1.462890625,
 15: 1.52880859375,
 16: 1.532928466796875,
 17: 1.5339584350585938,
 18: 1.534499168395996,
 19: 1.5346858501434326,
 20: 1.5347041813656688,
 21: 1.5347067745169625,
 22: 1.5347076659127197,
 23: 1.5347080319559154,
 24: 1.5347081039703099,
 25: 1.5347081124527746,
 26: 1.5347081143655024,
 27: 1.5347081150682849,
 28: 1.5347081152715,
 29: 1.5347081153022673,
 30: 1.534708115307436,
 31: 1.5347081153089157,
 32: 1.5347081153093955,
 33: 1.5347081153094975,
 34: 1.5347081153095137,
 35: 1.5347081153095172,
 36: 1.5347081153095183,
 37: 1.5347081153095186,
 38: 1.5347081153095186,
 39: 1.5347081153095186,
 40: 1.5347081153095186,
 41: 1.5347081153095186,
 42: 1.5347081153095186,
 43: 1.5347081153095186,
 44: 1.5347081153095186,
 45: 1.5347081153095186,
 46: 1.5347081153095186,
 47: 1.5347081153095186,
 48: 1.5347081153095186,
 49: 1.5347081153095186,
 50: 1.5347081153095186,
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 55: 1.5347081153095186,
 56: 1.5347081153095186,
 57: 1.5347081153095186,
 58: 1.5347081153095186,
 59: 1.5347081153095186,
 60: 1.5347081153095186,
 61: 1.5347081153095186,
 62: 1.5347081153095186,
 63: 1.5347081153095186,
 64: 1.5347081153095186,
 65: 1.5347081153095186,
 66: 1.5347081153095186,
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 68: 1.5347081153095186,
 69: 1.5347081153095186,
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 85: 1.5347081153095186,
 86: 1.5347081153095186,
 87: 1.5347081153095186,
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 114: 1.5347081153095186,
 115: 1.5347081153095186,
 116: 1.5347081153095186,
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 126: 1.5347081153095186,
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 128: 1.5347081153095186,
 129: 1.5347081153095186,
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 143: 1.5347081153095186,
 144: 1.5347081153095186,
 145: 1.5347081153095186,
 146: 1.5347081153095186,
 147: 1.5347081153095186,
 148: 1.5347081153095186,
 149: 1.5347081153095186,
 150: 1.5347081153095186,
 151: 1.5347081153095186,
 152: 1.5347081153095186,
 153: 1.5347081153095186,
 154: 1.5347081153095186,
 155: 1.5347081153095186,
 156: 1.5347081153095186,
 157: 1.5347081153095186,
 158: 1.5347081153095186,
 159: 1.5347081153095186,
 160: 1.5347081153095186,
 161: 1.5347081153095186,
 162: 1.5347081153095186,
 163: 1.5347081153095186,
 164: 1.5347081153095186,
 165: 1.5347081153095186,
 166: 1.5347081153095186,
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 169: 1.5347081153095186,
 170: 1.5347081153095186,
 171: 1.5347081153095186,
 172: 1.5347081153095186,
 173: 1.5347081153095186,
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 182: 1.5347081153095186,
 183: 1.5347081153095186,
 184: 1.5347081153095186,
 185: 1.5347081153095186,
 186: 1.5347081153095186,
 187: 1.5347081153095186,
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 189: 1.5347081153095186,
 190: 1.5347081153095186,
 191: 1.5347081153095186,
 192: 1.5347081153095186,
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 194: 1.5347081153095186,
 195: 1.5347081153095186,
 196: 1.5347081153095186,
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 198: 1.5347081153095186,
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 209: 1.5347081153095186,
 210: 1.5347081153095186,
 211: 1.5347081153095186,
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 213: 1.5347081153095186,
 214: 1.5347081153095186,
 215: 1.5347081153095186,
 216: 1.5347081153095186,
 217: 1.5347081153095186,
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 220: 1.5347081153095186,
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 223: 1.5347081153095186,
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 228: 1.5347081153095186,
 229: 1.5347081153095186,
 230: 1.5347081153095186,
 231: 1.5347081153095186,
 232: 1.5347081153095186,
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 234: 1.5347081153095186,
 235: 1.5347081153095186,
 236: 1.5347081153095186,
 237: 1.5347081153095186,
 238: 1.5347081153095186,
 239: 1.5347081153095186,
 240: 1.5347081153095186,
 241: 1.5347081153095186,
 242: 1.5347081153095186,
 243: 1.5347081153095186,
 244: 1.5347081153095186,
 245: 1.5347081153095186,
 246: 1.5347081153095186,
 247: 1.5347081153095186,
 248: 1.5347081153095186,
 249: 1.5347081153095186,
 250: 1.5347081153095186,
 251: 1.5347081153095186,
 252: 1.5347081153095186,
 253: 1.5347081153095186,
 254: 1.5347081153095186,
 255: 1.5347081153095186,
 256: 1.5347081153095186,
 257: 1.5347081153095186,
 258: 1.5347081153095186,
 259: 1.5347081153095186,
 260: 1.5347081153095186,
 261: 1.5347081153095186,
 262: 1.5347081153095186,
 263: 1.5347081153095186,
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 270: 1.5347081153095186,
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 289: 1.5347081153095186,
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 498: 1.5347081153095186,
 499: 1.5347081153095186,
 500: 1.5347081153095186}


sns.set()
fig = plt.figure(figsize = (12, 8))
ax = fig.add_subplot(xlim = (12, 50), ylim = (0, 1.65))


#3 Scatter Plots.
ax.scatter(x = list(ELs.keys()),
           y = list(ELs.values()),
           c = "#238A8D",
           lw = 3,
           label = "Expected Length")

ax.hlines(y = ELs[500],
           xmin = 12,
           xmax = 50,
           color = "#238A8D",
           ls = ':',
           lw = 2,
           label = "EL ≈ 1.534708115")

ax.scatter(x = list(space_p.keys()),
           y = list(space_p.values()),
           c = "#440154",
           lw = 3,
           label = "P(Char) = Space")

ax.hlines(y = 2/9,
           xmin = 12,
           xmax = 50,
           color = "#440154",
           ls = ':',
           lw = 2,
           label = "P = 2/9")

#Titles, Axes, & Legend
ax.set_title("How Long is the River?", fontsize = 24)
ax.set_xlabel("Character Location in Line", fontsize = 20)
ax.set_xticks([4*x for x in range(3, 13)])
ax.set_ylabel("Probability / Expected Length", fontsize = 18);
ax.tick_params('both', labelsize =  '20')
ax.legend(fontsize = '16', loc = 'center right')

fig.savefig('2025.05.23 FiddlerEC.png');

Setup¶

Constants and Functions¶

Rohan Lewis¶

2025.05.12¶