PROBABILITY HANDOUT



PROBABILITY HANDOUT

DEFN: A sample space, S, is a listing of all possible outcomes from an experiment.

Listed below are two examples of a sample space for an experiment consisting of flipping two fair coins:

S = {HH, HT, TH, TT}

S = {No heads, One head, Two heads}

DEFN: An event, A, is a subset of the sample space.

An experiment consisting of flipping two fair coins includes (but is not limited to) the following possible events:

A = getting two heads = {HH}

B = getting one head out of two flips = {HT, TH}

C = getting at least one head out of two flips = {HT, TH, HH}

DEFN: The complement of Event A is everything in the sample space that is not A. The complement of A is denoted [pic].

Notation: P(A) is read “the probability of event A’ and is the likelihood that event A will occur.

Facts: 0 ≤ P(A) ≤ 1

P(S) = 1

P(A) + P([pic]) = 1

Notation: P(A|B) is read “the probability of A given B” and is the conditional probability of event A occurring given that event B has occurred. This concept allows us to implement additional knowledge into our models.

DEFN: The union of events A and B is the set of all elements that occur in either A OR B (or both). The union of events A and B is denoted (A or B).

DEFN: The intersection of events A and B is the set of all elements that are common to both A AND B. The intersection of events A and B is denoted (A and B).

DEFN: Events A and B are said to be mutually exclusive if the occurrence of one events precludes the occurrence of the other event. If events A and B are mutually exclusive then: P(A and B) = 0; P(A|B) = 0; P(B|A) = 0.

DEFN: Events A and B are said to be independent if the occurrence of one event does not affect the probability of occurrence of the other event. If events A and B are independent then:

P(A|B) = P(A); P(B|A) = P(B).

RULES:

Additive Rule: P(A or B) = P(A) + P(B) – P(A and B)

Multiplicative Rule: P(A and B) = P(A) * P(B|A)

= P(B) * P(A|B)

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