TL;DR #1 is correct. #2: Quantity usually means just a number but isn't a maths term. A measures is a function that maps sets to numbers. A probability is a spe

TL;DR #1 is correct. #2: Quantity usually means just a number but isn't a maths term. A measures is a function that maps sets to numbers. A probability is a special kind of measure. For finite sets it is correct. More specifically, the quantity you are talking about is the number 1 and it is distributed over all elements of the set such that every element gets a number between 0 and 1 (where also the borders 0 or 1 are allowed), and the numbers of all elements sum up to 1. Quantity is not a mathematical term, but we usually use it in every day language to just denote a number. A measure, mathematically, is a function. A measure of a set A assigns _every subset_ of A to a number greater or equal 0 and has two key properties: 1) The empty set is always mapped to the number 0. 2) The union of every two subsets X and Y of A, that also don't have any element in common, is mapped to the sum of the values of the subsets X and Y, i.e. measure(X ∪ Y) = measure(X) + measure(Y). The cool thing is: Probabilities are a special kind of measures! Remember that a measure has to assign a number for every subset of a set A? Since A is a subset of itself, a measure has to also assign A a value. A so-called probability-measure is a measure that assigns A the value 1. Now you might guess where this is going: If you want to define probabilities for something going on in the real world, you first define a set that contains all the things that can happen. E.g. if you define probabilities for a dice roll, the set A contains the 6 possible numbers, A = {1,2,3,4,5,6}. Each subset of A corresponds to a possible event. For example, the set {2,5} means "either 2 or 5 appears". How is the probability-measure P defined? If the dice is fair, we have P({1})=P({2})=P({3})=P({4})=P({5})=P({6})=1/6. From property 2), we then get P({2,5})=1/6+1/6=1/3, i.e. the probability that 2 or 5 appear is 1/3. And of course we have P({1,2,3,4,5,6}) = 1, i.e. the probability that any number appears is 1. Sorry for the long reply, but I hope I could clarify a few things. Maths gets very fast very technical, but it's also nice to understand some key concepts.