TITLE: Intuition behind the Definition of Conditional Probability (for 2 Events)
QUESTION [27 upvotes]: What is some intuitive insight regarding the conditional probability definition: $P(A\mid B) = \large \frac{P(A \cap B)}{P(B)}$ ? I am looking for an intuitive motivation. My textbook merely gives a definition, but no true development of that definition. Hopefully that's not too much to ask.

REPLY [3 votes]: In summary, the essence of conditional probability is in every probability calculation, find the Whole (i.e. universal set), find the Part (intersection between the event you are trying to calculate the probability and the universal set) and divide the Part by the Whole to find the probability of given event (i.e the comparative relation between Part and Whole).
Let A be any given event and U be the set of all possible events or outcomes:
$ P(A)=\frac{P(A)}{P(U)} = \frac{P(A \cap U)}{P(U)} = P(A|U) $
If you are struggling to see the full equivalence above, remember that in set theory $ A \cap B = A $ when A is contained in B, which is always the case for any set when are trying to find the intersection with the Universal set (U).
Why do we need to divide?
Proportions: Finding the likelihood or probability of some event can be interpreted as a proportion problem, which implies the existence of a Part and a Whole. The way that you find how much a Part represent from the Whole is by dividing the Part by the Whole: $ \frac{Part}{Whole} $ (if you are still uncomfortable with the concept and for some reason can't visualize why division would provide you with such information, just take it as a rule for now, at some point the concept  will click). So whenever you are thinking about the probability or likelihood of an event, always think about the Part and the Whole, which leads to the next question: How do we find the Part and the Whole? We will need to understand how we represent events to answer that question.
Sets, Subsets and Cardinalities: An event is a set of outcomes, the cardinality of a set is the measure of the size of that same set. Now, for the sake of our probabilistic theory discussion, you will see that a Part is always a subset of the Whole (you may be thinking of some edge cases that do not fit this rule, keep reading and you will see that this idea indeed holds true for all cases).
The first step to find a probability of any event is to find the set that represents the Whole (i.e. the scope of all the possible outcomes, when you roll a die of 6 sides the Whole set can be represented as the numbers from 1 through 6, since they form all the possible outcomes), the second step is to find the Part (i.e. the set containing the subset of outcomes that you are trying to find the likelihood of occurrence).
Every probability calculation can be seen through the lens of a conditional probability (What!? Keep reading :) )
Let A be any given event and U be the set of all possible events or outcomes:
$ P(A)=\frac{P(A)}{P(U)} = \frac{P(A \cap U)}{P(U)} = P(A|U) $
If you are struggling to see the full equivalence above, remember that in set theory A and B = A when A is contained in B, which is always the case for any set when are trying to find the intersection with the Universal set.
As an example, say we are interested in knowing the probability of getting an even number when rolling a die of 6 sides, you can always formulate the problem thinking about conditional probability, when we defined that we are going to be rolling a die of six sides we just implicitly defined the Whole set (i.e the possible outcomes are numbers from 1 to 6), the second piece of information is the event itself that we would like to obtain the probability (even number), that is the Part. An important subtlety here, the Part you are interested here in the first place, is not ALL the even numbers but the ones that make part of your Whole set (i.e. subset of the Whole). How do you find that subset? The intersection of the two sets? precisely.
So the intersection gives you the Part that you are interested in, and once you have the Whole and Part you divide the two and find the probability (frequency of occurrence, or the comparative relation between the Part and the Whole).
If the outcomes or event that you are trying to find the likelihood does not make any part of the Whole,  in other words, if the intersection of both sets is empty then we know that the probability of such event is zero, the event is simply impossible to occur, there is no way you will get a 7 if you roll a die of 6 sides.
Always remember when you see P(A|B) we are already explicitly saying that B is the Whole or universal set, all you have to do now is to find the comparative relation between the Part and Whole..
To end, let me clarify a few common misconceptions:

On the context of conditional probability, aside from the relationship between the Whole and Part, the probability of what you are judging to be the Whole set does not matter by itself! As an example, the probability of someone having a tumor is irrelevant, if what we are trying to find is the probability of someone having a benign tumor given the person is already diagnosed with a tumor.

P(A|B) is not the same as P(B|A): The probability of someone having a benign tumor given the person is already diagnosed with a tumor, is not the same as the probability of someone being diagnosed with a tumor given the fact that the person has a benign tumor.

Shouldn't P(A|B) be equal to P(A and B)?
This is a common misinterpretation of what conditional probability means, we are not interested in the probability of A and B happening (i.e. $ \frac{P(A \cap B)}{P(U)} $), that's called joint probability and is a separate (and related) subject. Conditional probability means, now that B ALREADY HAPPENED, what is the probability of A also happening?