SET THEORY:
A set is a collection of things.
Each entry in a set is known as an element.
A union of two or more sets is another set that contains everything contained in the previous sets.
Union is designated by the symbol U.
If A and B are sets then A U B represents the union of A and B
The intersection of two (or more) sets is those elements that they have in common.
Intersection is designated by the symbol ∩.
So if A and B are sets then the intersection (the elements they both have in common) is denoted by A∩B.
PROBABILITY:
Probability is a measure of how likely is an event to happen.
It is measured in fractions from 0 to 1 (0 is impossible, 1 is unavoidable or certain).
Sometimes it is denoted in percentages, again from 0% to 100%.
Event is anything that happens. In probability theory we speak of events having outcomes or results.
Example flipping a coin (an event) has two possible outcomes—heads and tails.
When a coin is flipped (an event is tested), one of the outcomes is obtained.
If you know that the probability of an event (or one of the outcomes) is p, the probability of this event NOT happening (or the probability of it NOT having this given outcome), is (1-p).
p(not A) + p(A) = 1
If two (or more) independent events are occurring, and you know the probability of each, the probability of BOTH (or ALL) of them occurring together (event A and event B and event C etc) is a multiplication of their probabilities.
p(A and B) = p(A) * p(B)
p(A and B and C ... and Z) = p(A) * p(B) * p(C) * ... * p(Z)
If two (or more) incompatible events are occurring, the probability of EITHER of them occurring (event A or event B or event C etc) is a sum of their probabilities.
p(A or B) = p(A) + p(B)
p(A or B or C ... or Z) = p(A) + p(B) + ... + p(Z)
Incompatible means that they can't happen together, i.e. p(A and B) = 0. In case of two compatible events, the OR tool looks a bit more complicated:
p(A or B) = p(A) + p(B) - p(A and B)
If we know that A and B are independent, we can apply AND tool to rewrite:
p(A or B) = p(A) + p(B) - p(A) * p(B)
Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written P(A|B), and is read "the probability of A, given B". It is defined by
P(A|B)=P(A∩B)/P(B)