In more detail
P(X = k) = e^−λ × λ^k / k!, where λ is the average number of events per interval. The mean and the variance are both λ, which is a quick check on real data: if the variance is much larger than the mean, events are clustered and the Poisson model will understate extremes. With an average of 3 events per interval, exactly 2 occur with probability 0.2240. It models arrivals such as calls per hour, typos per page, server errors per minute or goals per match.
Why it matters
It answers staffing and capacity questions ("how often will more than 5 calls arrive in an hour?") and tells you whether a spike is unusual or just chance. It assumes events are independent and the rate is constant; rush hours, outages and contagion break that, so use it per time slot rather than across a whole day.
Example
A help desk averages λ = 3 calls per 10 minutes:
k 0 1 2 3 4
P(k) 0.0498 0.1494 0.2240 0.2240 0.1680
P(5 or more) = 1 − P(0 to 4) = 0.1847
So about one 10-minute slot in five and a half brings 5 or more calls, and about one in twenty brings none.
Common confusion
The Binomial distribution has a fixed number of trials. With many trials and a small chance, the two nearly agree: 1,000 trials at p = 0.003 give P(2) = 0.2242 binomially and 0.2240 by Poisson with λ = 3.