Two events are independent if the occurrence or non-occurrence of one does not change the probability that the other will occur. Example of Independent events: You roll 2 dice. Find the probability of rolling a 5 on each die. If events A and B are independent, then P (A ∩ B) = P (A). P (B)

1

Def: An experiment is a process by which observations are generated.

Def: A variable is a quantity that is observed in the experiment.

Def: The sample space (S) for an experiment is the set of all possible outcomes.

Def: An event E is a subset of a sample space. It provides the collection of outcomes that correspond to some classification.

Example:

Note: A sample space does not have to be finite.

Example: Pick any positive integer. The sample space is countably infinite.

A discrete sample space is one with a finite number of elements, { }1,2,3,4,5,6 or one that has a countably infinite number of elements { }1,3,5,7,… .

A continuous sample space consists of elements forming a continuum. { }x / 2 x 5< <

2

A Venn diagram is used to show relationships between events.

A intersection B = (A ∩ B) = A and B

The outcomes in (A intersection B) belong to set A as well as to set B.

A union B = (A U B) = A alone or B alone or both

Union Formula

For any events A, B, P (A or B) = P (A) + P (B) – P (A intersection B) i.e. P (A U B) = P (A) + P (B) – P (A ∩ B)

3

cA complement not A A ‘ A A = = = = A complement consists of all outcomes outside of A.

Note: P (not A) = 1 – P (A)

Def: Two events are mutually exclusive (disjoint, incompatible) if they do not intersect, i.e. if they do not occur at the same time. They have no outcomes in common.

When A and B are mutually exclusive, (A ∩ B) = null set = Ø, and P (A and B) = 0.

Thus, when A and B are mutually exclusive, P (A or B) = P (A) + P (B) (This is exactly the same statement as rule 3 below)

Axioms of Probability

Def: A probability function p is a rule for calculating the probability of an event. The function p satisfies 3 conditions:

1) 0 ≤ P (A) ≤1, for all events A in the sample space S

2) P (Sample Space S) = 1

3) If A, B, C are mutually exclusive events in the sample space S, then P(A B C) P(A) P(B) P(C)∪ ∪ = + +

4

The Classical Probability Concept: If there are n equally likely possibilities, of which one

must occur and s are regarded as successes, then the probability of success is s n

.

Example:

Frequency interpretation of Probability: The probability of an event E is the proportion of times the event occurs during a long run of repeated experiments.

Example:

Def: A set function assigns a non-negative value to a set.

Ex: N (A) is a set function whose value is the number of elements in A.

Def: An additive set function f is a function for which f (A U B) = f (A) + f (B) when A and B are mutually exclusive.

N (A) is an additive set function.

Ex: Toss 2 fair dice. Let A be the event that the sum on the two dice is 5. Let B be the event that the sum on the 2 dice is 6.

N(A) = 4 since A consists of (1,4), (2,3), (3,2), (4,1).

N (B) = 5 since B consists of (1,5), (2,4), (3,3), (4,2), (5,1)

N (A or B) = 4 + 5 = 9

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