I was given the following problem:
Let $X$ represent the difference between the number of heads and the number of tails obtained when a coin is tossed $n$ times. What are the possible values of $X$?
which already has an answer on this site. However, when I first approached this, I interpreted the word "difference" as the distance between two points lying on a horizontal/vertical line, and therefore always positive ($\vert 2n-h\vert,\,\,h=\{0,1,2,\ldots\}$). Similar questions on the lexical use of the word in math, agree that the definition of difference is given by Big number - small number. On the other hand, Wikipedia states that difference is the result of subtraction, and therefore may be negative. At this point, I guess this is left to the interpretation of the reader, since to me the question is ambiguously phrased.
Not content, I wanted to see if there existed any probability distributions with discrete support on $\mathbb{Z}$. I found out on the Wikipedia list that only $2$ are mentioned, namely the Degenerate distribution and Rademacher distribution. Every other distribution mentioned is defined on $\mathbb{N}$. So my question is, knowing this and what was mentioned in the paragraph above, wouldn't the best answer only include positive values of $X$? As most problems of this type involve Binomial distributions, Geometric distributions, etc. which are defined on $\mathbb{N}$.
On the other side, one benefit of considering negative values for $X$ is that it gives you more information: it tells you how many more tails or heads there are, depending on the sign of $X$.
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$\begingroup$My opinion of this matter is that the difference is always positive. This is because the problem used the phrase "difference BETWEEN". This is the same idea as the "distance" between two numbers on a number line, which is always positive.
Traditionally, the “difference" between two numbers refers to the distance on a number line between the points corresponding to each of the two numbers, a.k.a. the absolute value of their difference. No matter which one you subract from the other, the answer is always positive.
Let's denote the number of heads obtained when tossing a coin is X while the number of tails obtained when tossing the coin is Y.
Then the difference between the number of heads and the number of tails obtained when tossing a coin is denoted as |X - Y| or |Y - X|. The answer is always positive.
On some books, when you say "difference between a and b", it means a - b. My guess, and this is only my opinion, they do this because they want to promote an idea that most word problems that you can write in words can be translated to algebra in a simple way.
But when we say the difference between X and Y, it should always be positive.
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