Section 8.1 - Probability Models and Rules

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Section 8.1 - Probability Models and Rules

• Special Topics

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Definitions

• Random (not haphazard) A phenomenon or trial is
  said to be random if individual outcomes are
  uncertain but the long-term pattern of many
  individual outcomes is predictable.
• Randomness is a kind of order, an order that
  emerges only in the long run, over many
  repetitions.
• Examples hair color, the spread of epidemics,
  outcomes of games of chance, flipping coins, etc.

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Example Tossing a Coin

• Individual coin tosses are not predictable, so it
  would not be impossible to flip coins and see 5
  consecutive heads.
• However, if we are able to flip a coin
  indefinitely, we would see the true proportion of
  heads emerge, which is p .5. This is a
  long-run random probability.
• http//www.wiley.com/college/mat/gilbert139343/jav
  a/java04_s.html

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More Definitions

• Probability The probability of any outcome of a
  random phenomenon is the proportion of times the
  outcome would occur in a very long series of
  repetitions.
• Sample Space The sample space, or S of a
  random phenomenon is the set of all possible
  outcomes that cannot be broken down further into
  simpler components.
• Event An event is any outcome or any set of
  outcomes of a random phenomenon. That is, an
  event is a subset of the sample space.

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Examples

• Lets say we roll a die and flip a coin. Create
  the sample space to show all possible outcomes.
• S H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5,
  T6. There are 12 outcomes (2 x 6).
• A sample space that is unusually long can be
  truncated S H1, H2,T5, T6.

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Tree Diagram

• Another way to determine a sample space is with a
  tree diagram
• Thus, the sample space is S HH, HT, TH, TT.
• Note that there are 4 outcomes, but only 3
  events.
• This would be done on paper with symbols, not
  coins.

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Definitions Continued

• Probability Model A probability model is a
  mathematical description of a random phenomenon
  consisting of two parts a sample space S and a
  way of assigning probabilities to events.
• There are two ways to arrive at probabilities
• Empirical Probabilities These are probabilities
  arrived at through repeating an experiment, such
  as flipping a coin many times and recording the
  proportion of heads observed.
• Theoretical Probabilities These are
  probabilities arrived at through formulas and
  calculations.

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Still More Definitions

• Complement of an Event The complement of an
  event A is the event that A does not occur,
  written as AC.
• Disjoint Events Two events are disjoint events
  if they have no outcomes in common (they cant
  happen at the same time). Disjoint events are
  also called mutually exclusive events.
• Independent Events Two events are independent
  events if the occurrence of one event has no
  effect on the probability of the occurrence of
  the other event.
• Note! Independence and Disjoint (mutually
  exclusive) dont mean the same thing!

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Rules for Probabilities

• Any probability is a number between 0 and 1
  inclusive. So 0 P(E) 1. P(E) means
  probability of an event.
• All possible outcomes together must have
  probability of 1. This means that the sum of all
  the probabilities in a sample space equals 1.
• The probability that an event does not occur is 1
  minus the probability that the event does occur.
  This is saying that P(AC) 1 P(A).
• If two events are disjoint, the probability that
  one or the other occurs is the sum of their
  individual probabilities. The word or in
  probability means or add.

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Venn Diagram

• A Venn Diagram is a visual which helps to
  visualize probability situations. The following
  is a Venn Diagram for the Complement Rule.
• Thus, S A Ac. Recall that your sample space
  has a probability of 1, so A Ac 1.

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Another Venn Diagram

• This Venn Diagram illustrates the Rule for
  Disjoint Events
• Thus, the probability of A or B equals P(A)
  P(B).
• We will cover non-disjoint events tomorrow.

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Homework

• Worksheet 8.1