Unizor - Probability - Conditional - Problems 1

We recommend to attempt to solve these problems prior to listening to the lecture or reading answers and proofs provided.
Also assume that all probabilities mentioned in the problems are not equal to zero, that is we are excluding impossible events.

Problem 1.1.
Let A and B be two independent events defined in the sample space Ω.
Prove that the product of the probability of both events to occur by the probability of neither event to occur equals to a product of the probability of occurrence of only event A (but not B) by the probability of occurrence of only event B
(but not A).
In symbolic form using the set theory operations, prove that
P(A∩B)·P[Ω∖ (A∪B)] =
P(A∖ B)·P(B∖ A)
or, equivalently,
P(A∩B)·P[NOT (A∪B)] =
P[A∩(NOT B)]·P[B∩(NOT A)]

Problem 1.2.
Prove "geometrically" (see above) and probabilistically that if event A is independent of event B then event NOT A is also independent of event B.

Problem 1.3.
Prove "geometrically" and probabilistically that if event A is independent of event B then event A is also independent of event NOT B.

Problem 1.4.
Prove "geometrically" and probabilistically the following identity
P(A|B) + P[(NOT A)|B)] = 1

Problem 1.5.
Is the following equality always true (in other words, is it an identity)?
P(A|B) + P[A|(NOT B)] = 1