(c) Tobias Hossfeld (Aug 2021)
This script and the figures are part of the following book. The book is to be cited whenever the script is used (copyright CC BY-SA 4.0):
Tran-Gia, P. & Hossfeld, T. (2021). Performance Modeling and Analysis of Communication Networks - A Lecture Note. Würzburg University Press. https://doi.org/10.25972/WUP-978-3-95826-153-2
Penalty shootout We are looking at a penalty shootout between two teams: FC Bayern München and VFL Bochum. Every team has to shoot five penalties. FCB scores with a probability of 95%, while VFL scores with a probability of 85%. What is the joint probability that FCB scores $i$-times and VFL scores $j$-times. What is the probability that VFL scores more often than FCB?
import numpy as np import matplotlib.pyplot as plt from scipy.stats import binom n = 5 # number of penalties x = np.arange(n+1) p_fcb = binom.pmf(x, n, 0.95) # number of goals follows a binomial distribution p_vfl = binom.pmf(x, n, 0.85) plt.bar(x, p_fcb, width=-0.25, align='edge', color='r', label='FC Bayern') plt.bar(x, p_vfl, width=0.25, align='edge', label='VFL Bochum') plt.xlabel('score i') plt.ylabel('probability $P(X=i)$') plt.legend();
We are considering the joint probability mass function $P(X_1=i,X_2=j)$ that FC Bayern scores $i$ goals and VFL scores $j$ goals. The probabilities are independent of each other.
$ P(X_1=i,X_2=j) = P(X_1=i)\cdot P(X_2=j) $
from mpl_toolkits.mplot3d import Axes3D p_joint = p_fcb.reshape(6,1) @ p_vfl.reshape(1,6) fig = plt.figure() ax = fig.gca(projection = '3d') xpos, ypos = np.meshgrid(x, x) xpos = xpos.flatten('F') ypos = ypos.flatten('F') zpos = np.zeros_like(xpos) dx = 0.5 * np.ones_like(xpos) dy = dx.copy() dz = p_joint.flatten() ax.bar3d(xpos, ypos, zpos, dx, dy, dz) ax.view_init(elev=40, azim=230) plt.xlabel('goals FC Bayern ') plt.ylabel('goals VFL Bochum');
What is the probability that VFL Bochum wins: $P(X_1>X_2)$?