Content and Objective

Confirm results derived in the lecture when analyzing probability of sum of two dice being greater than 9, conditioned on the result of first dice being even and odd
Dice are sampled and occurences of according events are being counted

Import



In [1]:

    
# importing
import numpy as np

Simulation



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# number of trials
N_trials = int( 1e4 )

# counter of events "first dice even, first dice odd, sum of dice >= 9"
# counter of conditional events

# NOTE: Naming of events as on lecture slides

N_A = 0 
N_B = 0
N_C = 0

N_CA = 0
N_CB = 0


# loop for realizations
for _n in range( N_trials ):
    
    # sample 2 dice
    # NOTE: Upper limit is not included randint
    sample = np.random.randint( 1, 7, size=2 )
    
    # check for events; result: boolean
    first_even = ( sample[0] % 2 == 0 )
    sum_greater_9 = ( sum( sample ) >= 9 )
    
    # increase according counters
    
    ###
    # two methods for counting
    ###    

    # if you like "ifs" activate this one
    if 0:

        if first_even:
            N_A += 1
        else:
            N_B += 1

        if sum_greater_9:
            N_C += 1

            if first_even:
                N_CA += 1
            else:
                N_CB += 1
                
    # alternative solution    
    # as you can see N_CA, N_CB realize an "and" of events, so division by P(condition) has to be applied later on
    else:
        N_A += int( first_even )
        N_B += 1 - int( first_even )

        N_C += int( sum_greater_9 )

        N_CA += int( first_even ) * int( sum_greater_9 )
        N_CB += ( 1 - int( first_even ) ) * int( sum_greater_9 )
        

           
            
# print results         
print('Estimation of P(A):\t\t{}'.format( N_A / N_trials ))
print('Estimation of P(B):\t\t{}\n'.format( N_B / N_trials ))

print('Estimation of P(C|A):\t\t{}'.format( N_CA / N_A ))
print('Estimation of P(C|B):\t\t{}'.format( N_CB / N_B ))









    



Estimation of P(A):		0.5055
Estimation of P(B):		0.4945

Estimation of P(C|A):		0.333135509396637
Estimation of P(C|B):		0.21435793731041455



In [3]:

    
print('Estimation of P(C):\t\t{}'.format( N_C / N_trials ) )
print('P(C) by total probability:\t{}'.format( N_A / N_trials * N_CA / N_A + N_B / N_trials * N_CB / N_B ) )









    



Estimation of P(C):		0.2744
P(C) by total probability:	0.2744

Note:

Since we are dealing with Laplace experiments, we have $P(A)=|A|/|\Omega|$. So, conditional probabilities become $P(C|A)=\frac{\frac{|CA|}{|\Omega|}}{\frac{|A|}{|\Omega|}}=\frac{|CA|}{|A|}$.



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