## 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;

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