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Poisson Probability

Often one has to compute the probability of getting exactly n occurrences over a period of time or other continuum, for example:
  • a biologist wants to know if a patient's blood is normal by counting the red corpuscles in the squares of a ruled graticule applied on a slide containing diluted blood;
  • a quality control professional is asked to accept a batch of electronic components that the supplier guarantees as having at most 233 defectives per million;
  • an insurance claims officer assesses the likelihood of a home insurance holder having 1 or 2 fires over 5 years;
  • a highway engineer needs to know the probability of having more than 10 vehicles arriving at a tollgate in 1 minute;
  • a facilities manager wants to know if 2 switchboard operators can handle 95% of incoming calls;
  • a design engineer wants to establish the failure rate of a given product with a 95% confidence level;
  • a hospital manager needs to set up an emergency room knowing that the average rate of arrivals is 1.5 per 1 hour.
 
The Poisson distribution, named after the 19th century French mathematician Siméon Poisson, applies to such cases as these, where the events take place independently, at a known constant rate derived from a large sample (practically > 50), and with a low probability of occurrence (practically < 10%). The Poisson distribution is discrete, meaning that n can take only certain values, is defined by only 1 parameter (λ or mean), and mean and variance are equal (λ = σ²). For small values of p, Poisson is a good approximation for the binomial distribution.
 
The solution of the Poisson distribution function is given by the formula :
mathematical solution of the Poisson Probability function
For:
λ = np = mean;
n = random variable;
e = constant equal to approximately 2.718282.
 
Examples
 
(see and try areppim's Poisson Probability Calculator.
 
Example 1 : Patients arrive at a hospital emergency center at the rate of 3 per hour. What is the probability of two arrivals during the next 30 minutes?
 
λ = The arrival rate during 30 minutes = (3/60)*30 = 1.5.
n = The random variable = 2 occurrences.
 
The calculator produces the Poisson probabilities table as follows:
 
Number of occurrencesExact ProbabilityCumulative Probability
00.2230.223
10.3350.558
20.2510.809
 
The probability of exactly 2 arrivals in the next 30 minutes is 25.1%, and the probability of 2 or less arrivals is 80.9%.
 
Example 2 : Knowing that 3.5% of shoes from the total shop production are rejected, what is the probability of having to reject exactly 4 shoes in a batch of 90 shoes?
 
λ = The mean rejection rate in a batch of 90 = 90*3.5% = 3.15.
n = The random variable = 4 occurrences.
 
Number of occurrencesExact ProbabilityCumulative Probability
00.0430.043
10.1350.178
20.2130.390
30.2230.614
40.1760.789
 
The probability of having to reject 4 shoes is 17.6%.
 
Example 3 : The switchboard receives, on average, three telephone calls per minute. What capacity in terms of calls per minute should be installed to cope with incoming calls 95% of the time?
 
λ = Mean number of calls per minute = 3.
n = The random variable let us make it 10, in order to explore a wider distribution.
 
Number of occurrencesExact ProbabilityCumulative Probability
00.0500.050
10.1490.199
20.2240.423
30.2240.647
40.1680.815
50.1010.916
60.0500.966
70.0220.988
80.0080.996
90.0030.999
100.0011.000
 
The cumulative Poisson probability indicates that a switchboard of capacity 6 can handle incoming calls 96.6% of the time, i.e. a little better than the targeted 95%.