import numpy as np
import matplotlib
from matplotlib import pyplot as plt
import math

th = np.arange(0, 2*math.pi, 0.1)
prE3 = 2*(1-(math.sqrt(2)*(np.sin(th/2)**2))/(1+math.sqrt(2)))
matplotlib.rcParams.update({'font.size':10})
plt.figure(figsize = (5,3), dpi = 100)
plt.plot(th, prE3)
plt.xlabel(r'$\theta$')
plt.ylabel(r'$Pr(E_{3})$')
plt.show()

