The Poisson distribution describes the distribution of events which occur randomly in a continuous interval. e.g. Time, length, volume etc. The following conditions must apply:
 The events occur at random
 The events are independent of one another
 Two or more events do not occur simultaneously
 The probability that an event occurs in an interval is proportional to the size of that interval
The Poisson distribution has only one parameter and that is λ the average number of occurrences of an event in a given time interval. i.e. A rate.
Examples of Poisson Distribution
Examples of events which could follow a Poisson distribution are:
the number of cars visiting a service station in one half day.
the number of nails in a metre of wood.
the number of telephone calls arriving at a company's switchboard
Poisson Probability from Formula
The formula for Poisson probability is:
for x = 0, 1, 2, 3, ... λ is the mean number of occurrences in a given time or space and can be any positive value. 
Example
The average number of cars passing a speed camera in an hour, which are exceeding the speed limit is 5.
Find the probability that in the next hour 4 cars passing the camera will be speeding.
λ = 5 and x = 4
Poisson Probability from Tables
Rather than use the formula for finding Poisson probabilities, tables are available to look these up.
A table of the Poisson Distribution for values of λ from 0.1 to 6 are given with the Bursary examination.
The part of the table for for λ = 5 and x = 4 is shown below:
x\λ

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

5.8

6

0

0.015

0.0123

0.0101

0.0082

0.0067

0.0055

0.0045

0.0037

0.003

0.0025

1

0.063

0.054

0.0462

0.0395

0.0337

0.0287

0.0244

0.0207

0.0176

0.0149

2

0.1323

0.1188

0.1063

0.0948

0.0842

0.0746

0.0659

0.058

0.0509

0.0446

3

0.1852

0.1743

0.1631

0.1517

0.1404

0.1293

0.1185

0.1082

0.0985

0.0892

4

0.1944

0.1917

0.1875

0.182

0.1755

0.1681

0.16

0.1515

0.1528

0.1339

5

0.1633

0.1687

0.1725

0.1747

0.1755

0.1748

0.1728

0.1697

0.1656

0.1606

6

0.1143

0.1237

0.1323

0.1398

0.1462

0.1515

0.1555

0.1584

0.1601

0.1606

7

0.0686

0.0778

0.0869

0.0959

0.1044

0.1125

0.12

0.1267

0.1326

0.1377

If the above question asked for the probability that there are four or fewer speeding cars i.e. P(X ≤ 4) then the probabilities for x = 0, 1, 2, 3 and 4 would need to be added together.
P(X ≤ 4) = 0.0067 + 0.0337 + 0.0842 + 0.1404 + 0.1755 = 0.4405
Mean and Variance of the Poisson Distribution
There are formulae for working the mean, variance and standard deviation of a binomial distribution.
Mean (expected value)  μ = λ 
Variance 
σ^{2} = λ 
Standard deviation  σ = √λ 
In the example above:
E(X) = λ = 5
VAR(X) = λ = 5
SD(X) = √λ = √5 = 2.236 (to 4 sig. fig.)
Further Poisson Example
Cars arrive at a service station at the rate of 24 per hour. Let X represent the Poisson random variable for the number of cars arriving. a. Find the probability that 8 cars arrive in a 15 minute period. 
The probability that 8 cars arrive in 15 minutes is 0.1033 b. Find the probability that more than 3 cars arrive in a 15 minute period.
The probability that more that 3 cars arrive in 15 minutes is 0.8488 