Content and Objective

Simulation of probability of birthdays happening on the same day of the year (neglecting year of birth, leap years)
Birthdays of groups of according size are sampled and checked for doubles

Import



In [1]:

    
# importing
import numpy as np

import time

import matplotlib.pyplot as plt
import matplotlib

# showing figures inline
%matplotlib inline



In [2]:

    
# plotting options 
font = {'size'   : 20}
plt.rc('font', **font)
plt.rc('text', usetex=0)

matplotlib.rc('figure', figsize=(18, 6) )



In [3]:

    
start = time.time()

Parameters and Analytical Result



In [4]:

    
# max. group size in simulation and theory
N_max = 110

# group sizes used for simulation
N_sim = np.arange( 10, N_max, 10 )

# group sizes for theoretical results
N = np.arange( 1, N_max )

# theoretical value   
Pr_theo = np.zeros( len(N)  )

for ind_n, val_n in enumerate( N ):
    Pr_theo[ ind_n ] = 1 - np.prod( np.arange( 365. - val_n + 1 , 365 ) / 365 )

Simulation



In [5]:

    
# number of trials per group size
N_trials = int( 1e4 )

# initialize 
Pr_sim = np.zeros( len( N_sim ) )    

# loop for group sizes
for ind_n, val_n in enumerate( N_sim ):
    
    # initialize counters for multiple bithdays (hits) and number of trials (_n)
    hits = 0 

    # loop for realizations
    for _n in range( N_trials ):
        
        # get birthdays
        birthdays = np.random.randint(1, 366, size = val_n )

        # check double occurrences
        # NOTE: set( ) generates a set out of a list and thereby eliminates doubles
        if len( set( birthdays ) ) < val_n:
            hits += 1
            

    # estimate probability by relative frequency
    Pr_sim [ ind_n ] = hits / N_trials

Plotting



In [6]:

    
plt.plot( N, Pr_theo, linewidth=2.0, label='theo.')
plt.plot( N_sim, Pr_sim, 'x', mew=4, ms=18, linewidth=2.0, label='sim.')    

plt.xlabel('$N$')
plt.ylabel('$P($' + 'mind. 2 am gleichen Tag' +'$)$')
plt.grid( True )
plt.legend( loc = 'upper right' )









    Out[6]:





<matplotlib.legend.Legend at 0x7f56a9b152e8>



In [7]:

    
print(time.time()-start)









    



2.739107370376587