## Wednesday, April 1, 2015

### Unizor - Probability - Advanced Problems 4

The following problems are about conditional probability.
It is useful to represent events as certain closed sets of points on a plane (like a circle or a square or any other closed area of any, even irregular, shape).
Then a statement "event A has occurred" would be represented as "a point randomly thrown on a plane falls inside area A". The relationship "the occurrence of event B follows from the occurrence of event A" can be represented as an area A lying completely inside of area B. If a randomly thrown point falls inside A (that is, "event A has occurred"), it is automatically follows that this point is located inside B and event B can be considered as occurred.
A reminder about conditional probability of occurrence of event B if event A has occurred:
P(B|A) = P(B∩A)/P(A)
Graphically it can be represented as a big area Ω representing an entire sample space, areas A and B are inside it and intersect each other. In this case independence between A and B would be represented if the ratio of the area of their intersection A∩B towards the area of A equals to the ratio of the area of B to the area of Ω.

Problem A

Event X follows from event Y. True or false?
(a) event X follows from event NOT Y;
(b) event NOT X follows from event Y;
(c) event NOT X follows from event NOT Y;
(d) event Y follows from event X;
(e) event Y follows from event NOT X;
(f) event NOT Y follows from event X;
(g) event NOT Y follows from event NOT X;

Answer:
(a) false; (b) false; (c) false;
(d) false; (e) false; (f) false;
(g) true.

Problem B

Event X follows from event Y and also follows from event Z. True or false?
(a) event X follows from event Y AND Z;
(b) event X follows from event Y OR Z;
(c) event NOT Y follows from event (NOT X) OR (NOT Z);
(d) event NOT Y follows from event NOT (X OR Z);
(e) event NOT (Y AND Z) follows from event NOT X;
(f) event NOT (Y OR Z) follows from event NOT X;

Answer:
(a) true; (b) true; (c) false;
(d) true; (e) true; (f) true;

Problem C

Events X and Y are mutually exclusive.
(a) Are events X and Y independent?
(b) Are events X and NOT Y independent?
(c) Are events NOT X and NOT Y independent?

Answer:
(a) No; (b) No; (c) No.

Problem D

Given a standard deck of 52 cards ranking from 2 to 10, Jack, Queen, King and Ace in four different suits Spades, Hearts, Diamonds and Clubs.
A random experiment consists of pulling one card out of this deck.
Consider the following events:
E1: A card that is a Queen is pulled;
E2: A card that belongs to a suit of Spades is pulled;
E3: A card that belongs to a suit of Spades or Hearts is pulled;
E4: A Queen of Hearts is pulled;
(a) Are events E1 and E2 independent?
(b) Are events E1 and E3 independent?
(c) Are events E1 and E4 independent?

Answer:
(a) Yes; (b) Yes; (c) No;