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The Statistical Imagination
• Chapter 6. Probability Theory
and the Normal Probability
Distribution
© 2008 McGraw-Hill Higher Education
Probability Theory
• Probability theory is the analysis
and understanding of chance
occurrences
© 2008 McGraw-Hill Higher Education
What is a Probability?
• A probability is a specification of how
frequently a particular event of
interest is likely to occur over a large
number of trials
• Probability of success is the
probability of an event occurring
• Probability of failure is the probability
of an event not occurring
© 2008 McGraw-Hill Higher Education
The Basic Formula for
Calculating a Probability
• p [of success] = the number of
successes divided by the number
of trials or possible outcomes,
where p [of success] = the
probability of “the event of
interest"
© 2008 McGraw-Hill Higher Education
Basic Rules of Probability
Theory
• There are five basic rules of
probability that underlie all
calculations of probabilities
© 2008 McGraw-Hill Higher Education
Probability Rule 1: Probabilities
Always Range Between 0 and 1
• Since probabilities are proportions of a total number of
possible events, the lower limit is a proportion of zero
(or a percentage of 0%)
• A probability of zero means the event cannot happen,
e.g., p [of an individual making a free-standing leap of
30 feet into the air] = 0
• A probability of 1.00 (or 100%) means the event will
absolutely happen, e.g., p [that a raw egg will break if
struck with a hammer ] = 1.00
© 2008 McGraw-Hill Higher Education
Probability Rule 2: The Addition
Rule for Alternative Events
• An alternative event is where there is more
than one outcome that makes for success
• The addition rule states that the probability
of alternative events is equal to the sum of
the probabilities of the individual events
• For example, for a deck of 52 playing
cards:
p [ace or jack] = p [ace] + p [jack]
• The word or is a cue to add probabilities;
substitute a plus sign for the word or
© 2008 McGraw-Hill Higher Education
Probability Rule 3: Adjust for
Joint Occurrences
• Sometimes a single outcome is successful in more
than one way
• An example: What is the probability that a randomly
selected student in the class is male or single? A
single-male fits both criteria
• We call “single-male” a joint occurrence an event
that double counts success
• When calculating the probability of alternative
events, search for joint occurrences and subtract
the double counts
© 2008 McGraw-Hill Higher Education
Probability Rule 4: The
Multiplication Rule
• A compound event is a multiple-part event, such
as flipping a coin twice
• The multiplication rule states that the probability of
a compound event is equal to the multiple of the
probabilities of the separate parts of the event
• E.g., p [queen then jack] = p [queen] • p [jack]
• By multiplying, we extract the number of
successes in the numerator and the number of
possible outcomes in the denominator
© 2008 McGraw-Hill Higher Education
Probability Rule 5: Replacement
and Compound Events
• With compound events we must stipulate
whether replacement is to take place. For
example, in drawing a queen and then a
jack from a deck of cards, are we to
replace the queen before drawing for the
jack?
• The probability “with replacement” will
compute differently than “without
replacement”
© 2008 McGraw-Hill Higher Education
Using the Normal Curve as a
Probability Distribution
• With an interval/ratio variable that is
normally distributed, we can compute
Z-scores and use them to determine the
proportion of a population’s scores falling
between any two scores in the distribution
• Partitioning the normal curve refers to
computing Z-scores and using them to
determine any area under the curve
© 2008 McGraw-Hill Higher Education
Three Ways to Interpret the
Symbol, p
1. A distributional interpretation that describes
the result in relation to the distribution of
scores in a population or sample
2. A graphical interpretation that describes the
proportion of the area under a normal curve
3. A probabilistic interpretation that describes
the probability of a single random drawing
of a subject from this population
© 2008 McGraw-Hill Higher Education
Procedure for Partitioning Areas
Under the Normal Curve
1. Draw and label the normal curve
stipulating values of X and corresponding
values of Z
2. Identify and shade the target area ( p )
under the curve
3. Compute Z-scores
4. Locate a Z-score in column A of the
normal curve table
5. Obtain the probability ( p ) from either
column B or column C
© 2008 McGraw-Hill Higher Education
Information Provided in the
Normal Curve Table
• Column A contains Z-scores for one side of
the curve or the other
• Column B provides areas under the curve
( p ) from the mean of X to the Z-score in
column A
• Column C provides areas under the curve
from the Z-score in column A out into the
tail
© 2008 McGraw-Hill Higher Education
Important Considerations in
Partitioning Normal Curves
• The variable must be of interval/ratio
level of measurement
• The sample and population must be
normally distributed
• Always draw the curve and its target
area to avoid mistakes in reading the
normal curve table
© 2008 McGraw-Hill Higher Education
Percentiles and the
Normal Curve
• A percentile rank is the percentage of
a sample or population that falls at or
below a specified value of a variable
• If a distribution of scores is normal in
shape, then the normal curve and
Z-scores can be used to quickly
calculate percentile ranks
© 2008 McGraw-Hill Higher Education
Statistical Follies
• The Gambler’s Fallacy is the notion that past
gaming events, such as the roll of dice in the
casino game, Craps, are affected (or
dependent upon) past events
• E.g., It is fallacious to think that because a
coin came up heads three times in a row that
tails is bound to come up on the next toss
• The tosses are independent of one another
© 2008 McGraw-Hill Higher Education