# When asking a question like “What is the probability that our soccer team scores between 3 and 7 goals per game?”, does ‘between’ mean include or exclude the endpoints 3 and 7?

More generally, when working with a discrete random variable, X, (e.g. Poisson or Binomial) does the word ‘between’ mean include or exclude the endpoints?

**Answer:**

“What is the probability that *X* is *between* 3 and 7” is ambiguous with respect to the inclusion or exclusion of the endpoints when *X* is a discrete random variable.

There is NO consensus whether *‘between’* means include or exclude the endpoints. If you use the word *‘between’* with a discrete random variable then you have to explicitly tell the students whether you are *including* or *excluding* the endpoints.

Pr(3< *X *≤ 7) can be clearly and unambiguously written in words in a number of ways such as:

- What is the probability that
*X*is between 3 and 7, not including 3 but including 7? - What is the probability that
*X*is bigger than 3 but no bigger than 7? - What is the probability that
*X*is at least 4 but at most 7? - What is the probability that
*X*is greater than 3 but less than or equal to 7?

and so on.

Of course if *X* is a continuous random variable (Normal, uniform continuous or the triangular continuous) then the inclusion or exclusion of the endpoints is irrelevant and you may freely use the word ‘*between’* without clarifying whether the endpoints are included or excluded.

One more thing: don’t believe everything you read on Wolfram – it’s on a level similar to Wikipedia.