Download Probability Theory

The Statistical Imagination
• Chapter 6. Probability Theory and
the Normal Probability
Distribution
Probability Theory
• Probability theory is the analysis and
understanding of chance occurrences
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
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"
Basic Rules of Probability Theory
• There are five basic rules of probability that
underlie all calculations of probabilities
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 freestanding leap of 30 feet into the air] = 0
• A probability of 1.00 (or 100%) means that an
event will absolutely happen, e.g., p [that a raw
egg will break if struck with a hammer ]= 1.00
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
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 singlemale 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
Probability Rule 4: The
Multiplication Rule
• 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
• A compound event is a multiple-part event, such as
flipping a coin twice
• 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
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”
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
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
Procedure for Computing 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
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
Critical Z-scores
• Critical Z-scores are ones of great importance
in statistical procedures and are used very
frequently
• Some widely used critical Z-scores are 1.64,
1.96, 2.33, 2.58, 3.08, and 3.30
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