09-Probabilities

Event Probability from Poisson Distribution

Multiple Event Probability Estimates

Monte-Carlo

Example Monte-Carlo Simulation

#import needed packages
import numpy as np
import matplotlib.pyplot as plt

# Number of random numbers to generate
N = int(1e4)

MTBF_A = 1 # mean time between failures, unit time for Event A
muA = 1/MTBF_A # average events per unit time for A
1-np.exp(-muA) # probability of at least one event in unit time (poisson process)

0.6321205588285577

muA*np.exp(-muA) # probability of exactly one event in unit time (poisson process)

0.36787944117144233

unif = np.random.uniform(0, 1, N) # generate N random numbers between 0 and 1
# 1 if event, 0 if no event, where the if statement is the Monte Carlo approach
eventA = [1 if x < 1-np.exp(-muA)  else 0 for x in unif] 

Each entry in that list is an “event” over a duration of unit time and it can either be a failure or a success.

pA = sum(eventA)/len(eventA) # fraction of time with at least one event, probability of event in time unit
pA

0.6305

#determine the average rate of events per unit time
-np.log(1-pA)

0.9956045385938804

# now let's do the same thing for a Poisson process with a second 'event'
unif = np.random.uniform(0,1,N) #generate new random numbers
MTBF_B = 2 # mean time between failures
muB = 1/MTBF_B # average events per unit time
eventB = [ 1 if x < 1-np.exp(-muB) else 0 for x in unif ]

# now consider eventA and eventB as two independent Poisson processes in an OR gate
eventAorB = [min(1,sum([a,b])) for a,b in zip(eventA, eventB)]

pAorB = sum(eventAorB)/len(eventAorB) # fraction of time with at least one event
pAorB

0.7722

#combination of the two events for the average rate of events per unit time (mu)
-np.log(1-pAorB)

1.4792872279690552

# now consider eventA and eventB as two independent Poisson processes in an AND gate
eventAandB = [a*b for a,b in zip(eventA, eventB)]
pAandB = sum(eventAandB)/len(eventAandB) # fraction of time with at least one event
pAandB

0.2515

#Calculation of the probability of A and B occurring at the same time (product of the two probabilities)
pab = (1-np.exp(-muB))*(1-np.exp(-muA))
pab

0.2487200592643541

#combination of the two events for the average rate of events per unit time (mu) from simulation
-np.log(1-pAandB)

0.2896840751224541

#combination of the two events for the average rate of events per unit time (mu) from product of probabilities
-np.log(1-pab)

0.28597693937029134