import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

a = np.random.random_integers(1, 6, 4000)

a

allprobs = []
for i in range(300):
    a = np.random.random_integers(1, 6, 4000)
    allprobs.append(len([x for x in a.reshape(-1, 4) if 6 in x]) / 1000.0)

plt.hist(allprobs)

trials = 500
overall = []
for t in range(trials):
    initial = [20]
    time = 200
    for i in range(time):
        if np.random.random() > 0.5:
            initial.append(initial[-1] + 1)
        else:
            initial.append(initial[-1] - 1)
        if initial[-1] == 0:
            break
    if len(initial) < 201:
        initial += [0] * (201 - len(initial))
    overall.append(initial)


b = np.linspace(0, 200, 201)
for o in overall:
    plt.plot(b, o)

ends = []
for o in overall:
    ends.append(o[-1])

plt.hist(ends, 66)

import math

xsgood = []
ysgood = []
xsbad = []
ysbad = []
good = 0
trials = 5000
for i in range(trials):
    x = np.random.random()
    y = np.random.random()
    if (math.sqrt(x ** 2 + y ** 2) <= 1):
        good += 1
        xsgood.append(x)
        ysgood.append(y)
    else:
        xsbad.append(x)
        ysbad.append(y)
        
print(4 * float(good) / trials)

fig = plt.gcf()
ax = fig.add_subplot(111, aspect='equal')
plt.plot(xsgood, ysgood, "o")
plt.plot(xsbad, ysbad, "o")

endx = []
endy = []
trials = 10000
for i in range(trials):
    x = 0
    y = 0
    steps = 1000
    for i in range(steps):
        r = np.random.random_integers(1, 4)
        if r == 1:
            x += 1
        elif r == 2:
            x -= 1
        elif r == 3:
            y += 1
        elif r == 4:
            y -= 1
    endx.append(x)
    endy.append(y)

len(endx), len(endy)

plt.scatter(endx, endy)

x = [0]
y = [0]
steps = 10000
for i in range(steps):
    r = np.random.random_integers(1, 4)
    if r == 1:
        x.append(x[-1] + 1)
        y.append(y[-1])
    elif r == 2:
        x.append(x[-1] - 1)
        y.append(y[-1])
    elif r == 3:
        x.append(x[-1])
        y.append(y[-1] + 1)
    elif r == 4:
        x.append(x[-1])
        y.append(y[-1] - 1)
plt.plot(x, y)
plt.plot(0, 0, "go")
plt.plot(x[-1], y[-1], "ro")