What are rare events? That is an idea posed to Stats students frequently, that sometimes leads to confusion. Speaking generally, a rare event is an event that is very unlikely to happen, an event that has small likelihood of occurring. And such likelihood is measured as a probability. So then, in other words, a rare event is simply an event with a small probability of occurrence

How Small is Small?

That is a good question: How small the probability of the occurrence of a certain event must be in order for us to call it a rare event? The answer is: It depends. The probability threshold needs to be pre-specified before we can call an event to be rare. The typical threshold used in most Statistics courses is 0.05. So, an event will be rare if its probability of occurrence is less than 0.05.

How to Write it Mathematically?

Let A be a probability event (let us recall that a probability space is a subset of the sample space $\Omega$, which corresponds to all possible outcomes of a probability experiment). We say the event A is rare, or unusual , if

That is it. Simple. You are given an event, you compute its probability, if it is less than 0.05 (or whatever the pre-specified threshold for unusual events is), then it is considered to be rare or unusual, otherwise, it is a usual event.

What Complications Can I Find?

Not many complications should be found with the concept of rare event itself. Typically, the hardest part could to compute the probability of a certain event (which can always be tricky....do not forget, calculating probabilities is not always easy). Once you know the probability of the event, you simply check if it is less than 0.05. Make sure that 0.05 is actually the threshold that is understood for unusual events. In fact, when not specified you can safely assume it is 0.05.

Example

A family has 6 children, and all of them are men. Can that situation be considered as a rare event?

Answer : Let X be the number of men out of the six children. Based on the information provided, we have that X has a Binomial Distribution, with parameters n = 6 and p = 0.5. We need to compute the following probability:

$$\Pr \left( X = 6 \right)={{C}_{6, 6} \times {0.5}^{6}}\times {{\left( 1-{0.5} \right)}^{6-6}}=1\times {0.5}^{6}\times {0.5}^{0}= 0.0156$$

Since the probability of the event is 0.0156, which is less than 0.05, this event is considered to be a rare or unusual event.