Exam 1

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Exam 1 115a
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Basic Probability

• For any event E,
• The union of two sets A and B, A ? B, includes
  items that are in either A or B.
• The intersection, A ? B, is the set of all items
  in both A and B.
• The union of two sets can be found by

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Basic Probability (cont.)

• If E and F are mutually exclusive,
• then
• and because
• The complement of E, EC, is the event that E does
  not happen
• so
• and , the sample space

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Basic Probability (cont.)

• DeMorgans Laws

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Summation Notation

• Summation notation is used to write series
  efficiently
• The letter k is called the index of summation
• The numbers 1 and n are the lower limit and upper
  limit of the summation, respectively

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Summation Properties
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Summation Properties
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Random Variables

• We may assign a number, X, to every event that
  arises from an experiment.
• Random variables only take on numerical values!
• Given an experiment, a random variable, X, and a
  number, x, the expression Xx stands for the
  event that an outcome occurs to which X is
  assigned the value x.

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Random Variables (cont.)

• We may then speak of probabilities of these
  events, i.e. the probability that the event Xx
  is P(Xx).
• The sum of the probabilities for all possible
  values of a random variable is equal to 1 (i.e.
  you have accounted for all possible outcomes of
  the sample space).

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Expected Value

• The expected value of X, denoted E(X), is the sum
  of all values of X, weighted by their respective
  probabilities. That is
• If the experiment is repeated a large number of
  times, and the value of X is noted for each
  trial, then the average of these observed values
  will approach E(X).
• Know the interpretations!!!!

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Expected Value (cont.)

• It is usually convenient to set up a chart for
  all values of X. Each row corresponds to each
  value, the last column is the product of the
  previous 2 columns, and the expected value of X
  is the sum of this last column.

x P(Xx) x.P(Xx)


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Conditional Probability

• The conditional probability of an event E, given
  that an event F has happened, is denoted P(EF).
• If , then
• Note, generally.

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Independent Events

• Sometimes knowing that F has happened does not
  change the probability of E, that is,
  . We then say that E and F are
  independent of each other.
• Tests for independent events, E and F

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Independent Events (cont.)

• If E, F, and G are three events, and E and F are
  independent given that G has happened, then
• this may be generalized to include more
  . than 2 independent events

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Bayes Theorem

• If the sets B1, B2, , Bn partition the sample
  space, S, and A is any event, then
• Also, for any k with ,

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Computer Tools

• Microsoft Word for word processing mathematics
• know how to open the equation editor
• know the equation editor window

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Computer Tools (cont.)

• Microsoft Excel
• understand the layout of a spreadsheet (for
  example, you should know what cell I am referring
  to if I specify cell B11)
• know how to insert a formula
• know the database functions
• DCOUNT
• DAVERAGE
• DMAX

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Computer Tools (cont.)

• Microsoft Excel (cont.)
• with each of those functions, you should
  recognize
• what is a valid input for the function
• the dialogue box that appears
• how to indicate the input fields
• what fields to leave blank (i.e. in DCOUNT)

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Exam 1

• Combination of multiple choice questions and
  mostly short answer problems
• Scratch paper is available, so the only required
  items for you to bring are calculators and
  pencils
• NO CELL PHONES!!!!! If any cell phones ring or
  buzz during the exam, you may lose credit on the
  exam.

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Formula Sheet