Introducing Probability

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Chapter 10

• Introducing Probability

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Idea of Probability

• Probability is the science of chance behavior
• Chance behavior is unpredictable in the short
  run, but is predictable in the long run
• The probability of an event is its expected
  proportion in an infinite series of repetitions

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How Probability BehavesCoin Toss Example
Eventually, the proportion of heads approaches 0.5
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How Probability BehavesRandom number table
example
The probability of a 0 in Table B is 1 in 10
(.10) Q What proportion of the first 50 digits
in Table B is a 0? A 3 of 50, or 0.06 Q
Shouldnt it be 0.10? A No. The run is too short
to determine probability. (Probability is the
proportion in an infinite series.)
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Probability Models

• Probability models consist of two parts
• Sample Space (S) the set of all possible
  outcomes of a random process.
• Probabilities for each possible outcome in sample
  space S are listed.

Probability Model toss a fair coin S Head,
Tail Pr(heads) 0.5 Pr(tails) 0.5
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Rules of Probability
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Rule 1 (Possible Probabilities)

• Let A event A
• 0 Pr(A) 1
• Probabilities are always between 0 and 1.
• Examples
• Pr(A) 0 means A never occurs
• Pr(A) 1 means A always occurs
• Pr(A) .25 means A occurs 25 of the time

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Rule 2 (Sample Space)

• Let S the entire Sample Space
• Pr(S) 1
• All probabilities in the sample space together
  must sum to 1 exactly.
• Example Probability Model toss a fair coin,
  shows that Pr(heads) Pr(tails) 0.5 0.5 1.0

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Rule 3 (Complements)

• Let A the complement of event A
• Pr(A) 1 Pr(A)
• A complement of an event is its opposite
• For example
• Let A survival ? then A death
• If Pr(A) 0.95, then
• Pr(A) 1 0.95 0.05

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Rule 4 (Disjoint events)

• Events A and B are disjoint if they are mutually
  exclusive. When events are disjoint
• Pr(A or B) Pr(A) Pr(B)
• Age of mother at first birth
• (A) under 20 25
• (B) 20-24 33
• (C) 25 42

Pr(B or C) 33 42 75
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Discrete Random Variables
Discrete random variables address outcomes that
take on only discrete (integer) values
Example A couple wants three children. Let X
the number of girls they will have This
probability model is discrete
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Continuous Random Variables
Continuous random variables form a continuum of
possible outcomes.

• Example Generate random number between 0 and 1 ?
  infinite possibilities.
• To assign probabilities for continuous random
  variables ? density models (recall Ch 3)

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Area Under Curve (AUC)

• The AUC concept (Chapter 3) is essential to
  working with continuous random variables.

Example Select a number between 0 and 1 at
random. Let X the random value. Pr(X lt .5)
.5 Pr(X gt 0.8) .2
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Normal Density Curves
Introduced in Ch 3 XN(µ, ?).
?
? Height XN(64.5, 2.5)
Standardized ZN(0, 1)
Z Scores
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68-95-99.7 Rule

• Let X ? height (inches)
• X N (64.5, 2.5)
• Use 68-95-99.7 rule to determine heights for
  99.7 of ?
• µ 3s 64.5 3(2.5)
• 64.5 7.5 57 to 72

If I select a woman at random ? a 99.7 chance
she is between 57" and 72"
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Calculating Normal Probabilities when 68-95-99.7
rule does not apply

• Recall 4 step procedure (Ch 3)
• A State
• B Standardize
• C Sketch
• D Table A

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Illustration Normal Probabilities
What is the probability a woman is between 68
and 70 tall? Recall X N (64.5, 2.5)
A State We are looking for Pr(68 lt X lt 70)
B Standardize
Thus, Pr(68 lt X lt 70) Pr(1.4 lt Z lt 2.2)
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Illustration (cont.)
C Sketch
D Table A Pr(1.4 lt Z lt 2.2) Pr(Z lt 2.2) -
Pr(Z lt 1.4) 0.9861 - 0.9192 0.0669