import matplotlib.pyplot as plt
import matplotlib.ticker as mtick
import numpy as np
import pandas as pd
from pandas.api.types import CategoricalDtype
import seaborn as sns

#Creates an nXn rhombus with all probabilities, probability of knocking down next pin is p.
def probRhombus(n, p) :
    
    #list of lists of probabilities
    rhombus = [[1]]
    
    #n-1 steps ascending
    asc = n-1
    for a in range(asc) :
        #Append 0's to beginning and end of list for consistency.
        new = getProbs([0] + rhombus[a] + [0], p)
        rhombus.append(new)
    
    #n-1 steps descending.
    desc = n-1
    for d in range(desc):
        new = getProbs(rhombus[asc+d], p)
        rhombus.append(new)
        
    return(rhombus)


#Takes a list of n probabilities and spits out a list of n-1 probabilities. 
def getProbs(probs, p) :
    
    newprobs = []
    for i in range(len(probs)-1) :
        #inclusion exclusion
        newprob = p*probs[i] + p*probs[i+1] - p*p*probs[i]*probs[i+1]
        newprobs.append(newprob)
    return(newprobs)

ns = [2, 5, 10, 20, 50, 100, "∞"]
probabilities = [x/400 for x in range(200,400)]

final_pins = []

for n in ns:
    for p in probabilities :
        
        if n == "∞" :
            final_pins.append(((2*p-1) / p**2))
        else :
            r = probRhombus(n, p)
            final_pins.append(r[2*n-2][0])

data = {'Dimension': np.repeat(np.array(ns), [200]),
        'KD': probabilities * 7,
        'FP': final_pins}

df = pd.DataFrame(data)
df['Dimension'] = df['Dimension'].astype(CategoricalDtype(categories = ['2', '5', '10', '20', '50', '100', "∞"],
                                                          ordered=True))

sns.set()

#Plot.    
fig = plt.figure(figsize = (13, 8))
ax = fig.add_subplot()

scatter = plt.scatter(x = df['KD'], y = df['FP'], s = 35, c = df['Dimension'].cat.codes, cmap = 'viridis_r')
                                 
ax.set_title("End Behavior of N^2 Bowling", fontsize = 24)

ax.set_xlabel("Probability of Knocking Down Next Pin", fontsize = 22)
ax.xaxis.set_major_formatter(mtick.PercentFormatter(1))

ax.set_ylabel("Probability of Knocking Down Final Pin", fontsize = 22)
ax.xaxis.set_major_formatter(mtick.PercentFormatter(1))
ax.tick_params(axis = 'both', labelsize = 18)

plt.legend(title = "   Rhombus\nDimension (N)",
           title_fontsize = 20,
           handles = scatter.legend_elements()[0],
           labels = ns,
           ncol = 2,
           loc = "lower right",
           fontsize = 18)

plt.savefig("2022.08.05 Classic.png");

Rohan Lewis

2022.08.05