Probability of Simple Events Section 5'1

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Probability of Simple EventsSection 5.1

• Alan Craig
• 770-274-5242
• acraig_at_gpc.edu

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Objectives 5.1

• Understand the properties of probabilities
• Compute and interpret probabilities using the
  classical method
• Compute and interpret probabilities using the
  empirical method
• Use simulation to obtain probabilities
• Understand subjective probabilities

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Why Study Probability?

• Statistical measures are based on probability.
• To understand and calculate statistics, we need
  to know some basic probability.

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Probability

• Definition
• Measure of the likelihood of a random phenomenon
  or chance behavior
• The long-term proportion with which a certain
  outcome is observed is the probability of that
  outcome.

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Probability

• The probability of getting heads when flipping
  a fair coin is 50 or 0.5
• However, if we flip a coin two times, we could
  easily get heads both times.
• If we flip a coin two thousand times, we should
  see the outcome heads about ½ the time.

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Law of Large Numbers

• As the number of repetitions of a probability
  experiment increases, the proportion with which a
  certain outcome is observed gets closer to the
  probability of the outcome.

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Experiment

• Any process that can be repeated in which the
  results are uncertain.
• Lets do an experiment!
• Everyone flip a coin three times and record heads
  or tails.

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Experiment

• Did anyone get three heads? Three tails?
• How many heads altogether?
• How many tails altogether?

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Events and Sample Spaces

• Simple Event Any single outcome from a
  probability experiment. Each simple event is
  labeled ei
• Sample Space S Set of all possible simple
  events.
• Event Any set of outcomes from a probability
  experiment. Labeled E

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Example 1, p. 187 (a) Simple Events for Rolling a
Die
e1 rolling a one 1 e2 rolling a two
2 e3 rolling a three 3 e4 rolling a
four 4 e5 rolling a five 5 e6 rolling
a six 6
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Example 1, p. 187 (b) Sample Space for Rolling a
Die
Sample Space e1 U e2 U e3 U e4 U e5 U e6
1,2,3,4,5,6 The union of all simple events.
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Example 1, p. 187 (c) Event E, Roll an Even Number
Event E, roll an even number e2 U e4 U e6
2,4,6 The union of all simple events that
comprise the Event E.
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Properties of Probabilities

• P(E) probability of an event likelihood of
  that event occurring
• 0 P(E) 1 (e.g., 0, .06, .37, .5, .83, 1)
• Impossible event ? P(E) 0
• Certain event ? P(E) 1
• P(S) P(e1) P(e2) P(en) 1

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Properties of Probabilities

• Unusual event an event that has a low
  probability of occurring
• In statistics we will use 5 most of the time as
  the cut off for unusual events.
• Sometimes statisticians use 10 or 1 depending
  on the circumstances

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Classical Method

• If an experiment has n equally likely simple
  events and if the number of ways that an event E
  can occur is m, then P(E), the probability of E,
  is
• N(E) is the number of simple events in E and N(S)
  is the number of simple events in the sample
  space S.

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Example Classical Method

• Using Figure 2 on p. 190, what is S and N(S)?
• Find the probability of rolling a total of 4 with
  a pair of dice.

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Example Classical Method

• Using Figure 2 on p. 190, what is S and N(S)?
• S (1,1), (1,2), (1,3), (1,4), (1,5),
  (1,6),
• (2,1), (2,2), (2,3), (2,4),
  (2,5), (2,6),
• (3,1), (3,2), (3,3), (3,4),
  (3,5), (3,6),
• (4,1), (4,2), (4,3), (4,4),
  (4,5), (4,6),
• (5,1), (5,2), (5,3), (5,4),
  (5,5), (5,6),
• (6,1), (6,2), (6,3), (6,4),
  (6,5), (6,6)
• N(S) 36

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Example Classical Method

• Find the probability of rolling a total of 4 with
  a pair of dice. First find E.
• S (1,1), (1,2), (1,3), (1,4), (1,5),
  (1,6),
• (2,1), (2,2), (2,3), (2,4),
  (2,5), (2,6),
• (3,1), (3,2), (3,3), (3,4),
  (3,5), (3,6),
• (4,1), (4,2), (4,3), (4,4),
  (4,5), (4,6),
• (5,1), (5,2), (5,3), (5,4),
  (5,5), (5,6),
• (6,1), (6,2), (6,3), (6,4),
  (6,5), (6,6)

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Example Classical Method

• Find the probability of rolling a total of 4 with
  a pair of dice.
• E (1,3), (2,2), (3,1) ? N(E) 3

• Is rolling a total of 4 with a pair of dice an
  unusual event?

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Tree Diagram

• A way to illustrate classical probability is with
  a tree diagram
• We flip a coin 3 times. Draw a tree diagram to
  list the equally likely outcomes of this
  experiment.

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Tree Diagram
Flip 1
T
H
Flip 2
Flip 2
H
T
T
H
Flip 3
Flip 3
Flip 3
Flip 3
H
T
H
T
T
H
H
T
H H H
H H T
T H H
H T H
T H T
H T T
T T H
T T T
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Empirical Probabilities

• The empirical probability of an event E is
  approximately the number of times an event E is
  observed, divided by the number of repetitions of
  the experiment (e.g. number of survey
  respondents).

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Empirical Probabilities

• What is the probability of shoplifting in the
  random sample in Problem 20, p. 197?
• What is the probability of purse snatching?
• Is either a statistically unusual event?

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Empirical Probabilities

• What is the probability of shoplifting in the
  random sample in Problem 20, p. 197?
• 118 / (55118197774310545)
• 118 / 595
• 0.198
• Not statistically unusual event (0.198 gt .05)

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Empirical Probabilities

• What is the probability of purse snatching in the
  random sample in Problem 20, p. 197?
• 5 / 595
• .008
• Purse snatching is a statistically unusual event
  (0.008 lt .05)

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Simulation and Subjective Probabilities

• We can simulate an experiment without actually
  conducting it by using a computer or calculator
  to generate the events. We use the random number
  generator to do this.
• Subjective probabilities are educated guesses by
  experts. But people interpret data differently,
  so use extreme skepticism with subjective
  probabilities.

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Questions

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