Basic probability: chance of observing sets from draws w/wo replacement

Question

As will be evident from the question I am a complete novice in probability. I am trying to figure out which probability domain I need to look for in questions as: If one have a finite number of the labels, say 100 different ones then what will be the probability of drawing a certain set of labels (the order does not matter) given D draws. I suspect that if you have infinite copies of each label it is like drawing and putting the label back into the pool. For instance how would one start to answer:

1. What is the chance of getting two similar labels from eight draws.
2. Extend that to, say, 3 times 3 similar labels and 5 times 2 similar labels from 1000 draws.
3. What is the probabilities for the above two questions if you only have 100 copies of each label (and thus no replacement)

I would happily appreciate suggestions to further online reading of what I know is very basic probability theory.

Answer 1

The theory you need to approach these kind of problems is essentially counting. You have to identify all the possible outcomes of your experiment, defining the outcomes in a way so that each outcome has the same chance of occurring. You count all possible outcomes, and you also count all outcomes that lead to the condition you are interested to be satisfied. Then you divide latter by the former and you get your probability.

Note that the way of defining the outcomes might not always be the most straight forward one, since you need each outcome to have the same chance of occurring. This is guaranteed if you have symmetry between the outcomes. For example, if you choose repeatedly from an urn with balls that have labels, the labels are repeated, and some appear more often than others, you cannot take the outcomes to be a sequence of those labels. Instead, you have to give each ball a new unique label and define the outcomes accordingly.

Once you know what to count, the theory of mathematics that can be useful in giving you the tools to do so is called combinatorics.

All you need for this kind of problems is finite discrete probability theory and if the counting gets difficult: combinatorics. Most introductions to probability will cover this. A resource I certainly recommend is the Khan Academy probability course. The first couple of lectures will be sufficient for this purpose.