Unizor - Probability Definition - Problems 3

Problem A

Consider two events, X and Y (not necessarily mutually exclusive).
Prove that
P(X∪Y) ≤ P(X) + P(Y).

Problem B

Let A be an event and A be its negation (that is, NOT A).
Prove that
P(A) + P(A) = 1.

Problem C

Consider two events, X and Y.
It is given that
P(X∪Y) = P(X∩Y)
Prove that events X and Y are essentially identical, that is they contain exactly the same elementary events with positive probabilities, while, if there are elementary events with zero probability, they might constitute the difference between X and Y.

Problem D

The random experiment consists of flipping a coin two times in a row.
(a) represent all elementary events in this experiment as strings of letters H (for "heads") and T (for "tails"). What are their probabilities?
(b) using the symbolics from the previous problem (a), express the events E1 - "No tails resulted", E2 - "Only one head resulted", E3 - "Number of heads not equal to number of tails" and E4 - "Number of heads equal to number of tails". What are the probabilities of these events?
(c) Which elementary events constitute the events E1 OR E2 and E3 OR E4. What are the probabilities of these events?
(d) Which elementary events constitute the events E2 AND E4 and E1 AND E3. What are the probabilities of these events?
(e) Which elementary events constitute the events NOT E2 and NOT E4. What are the probabilities of these events?