## Glossary of terms |

You are here: areppim > Glossary of terms > Poisson Probability Function

**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 :

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 occurrences Exact Probability Cumulative Probability 0 0.223 0.223 1 0.335 0.558 2 0.251 0.809 - 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 occurrences Exact Probability Cumulative Probability 0 0.043 0.043 1 0.135 0.178 2 0.213 0.390 3 0.223 0.614 4 0.176 0.789 - 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 occurrences Exact Probability Cumulative Probability 0 0.050 0.050 1 0.149 0.199 2 0.224 0.423 3 0.224 0.647 4 0.168 0.815 5 0.101 0.916 6 0.050 0.966 7 0.022 0.988 8 0.008 0.996 9 0.003 0.999 10 0.001 1.000 - 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%.