Probability

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Probability The Relative Frequency Definition

• We define probability as a relative frequency.
• If some event A occurs f times out of N possible
  opportunities, we say that the probability of A
  is f divided by N

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Probability experiments

• We usually describe these as probability
  experiments.
• Thus if an experiment is repeated enough, say N
  times, and if event A occurs f times, then the
  probability of A is

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Intuitive Definition

• Note that probability is an intuitive concept.
• It is both analytic and empirical.
• We theorize about probability analytically, and
  hence use formal proof.
• We think about it and demonstrate it with real
  world observation.

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Some Fundamentals

• The probability of an impossible event must be
  zero (0.0)
• The probability of a certain event must be 1.0
• Any and all probabilities must be between 0.0 and
  1.0

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Calculating Event probabilities

• The probability that an event will occur is the
  sum of the sub-events that make comprise the
  event.

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Composite Events

• A composite event is one that comprised of
  combinations of other events
• Such as
• If either of two events occur
• If both of two events occur

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Intersection

• Think in terms of Venn diagrams
• Picture two sets A and B
• What is the probability of being both A and or B?

A
B
C
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The Addition Rule

• If A and B are two events, and their
  probabilities are denoted by P(A) and P(B), then
  the probability of either A or B or both is
  denoted by P(A or B).
• P(A or B) P(A) P(B) - P(A and B)

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The Addition Rule - a simple example

• What is the probability of getting a head on
  either of two tosses of a coin?
• P(A) ½
• P(B) ½
• P(A and B)1/2 ½ ¼
• P(A or B) P(A) P(B) - P(A and B)
• P(A or B) ½ ½ - ¼ ¾

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Mutually Exclusive events

• If two events are mutually exclusive, they cannot
  both occur
• (e.g. both a 2 and a 3 on a single role of a die)
• The addition rule for mutually exclusive events
  is
• P(A or B) P(A) P(B)

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The Complement of an event

• The complement of an event occurs when the event
  itself does not occur
• P(A) P(A) 1.0
• P(A) 1.0 - P(A)

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Joint Probabilities

• A joint probability is the probability that both
  A and B will occur P(A and B)

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Example of Joint Probabilities
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Marginal Probabilities
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Conditional Probabilities
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The Multiplication Rule
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Statistical Independence
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