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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 :

For:

λ = np = mean;

n = random variable;

e = constant equal to approximately 2.718282. ## Examples

- Check 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%.