CHAPTER SIX:
Probability
In the social sciences, researchers speak of
likelihood; not proof. The study of abstract constructs
does not lend itself to direct manipulation or observation.
Likelihood of abstract events can be determined by
assessing the probability of the event. PROBABILITY can be
expressed as a proportion. If an event can only turn out
two ways, and only one of those ways is acceptable, the
chances of an acceptable outcome are 1 in 2. Expressed as a
fraction, this becomes 1/2: the numerator is the number of
acceptable outcomes and the denominator is the number of
outcomes, acceptable and unacceptable.
Consider a die (one of a pair of dice) with 6 equal
sides. Each side has a different number, 1 through 6. If
any even number will do, the chance of a success is 3 in 6,
or 3/6. This reduces to 1/2, which can be expressed as 5050. The chances of it going either way (success or failure
to get what was wanted) are 50%.
LIKELIHOOD AS AREA:
If likelihood can be expressed as a fraction, or
proportion, than likelihood can be depicted as the
proportion of area under the curve of a polygon. If any
score less than the mean of a distribution is acceptable,
than the chances of getting an acceptable score is 50% and
can be depicted as the lower half of a polygon:
The likelihood of an event can be determined when all
possible outcomes are known and then used to form a
frequency distribution of all of those potential outcomes,
presented as a polygon. If a distribution is normally
distributed, the area under the curve can be determined
with a calculus derivative. This has already been done for
Z-scores between -4 to +4 in the UNIT NORMAL TABLE. Divided
into three columns, column A identifies the Z Score, column
B identifies all the area below the Z score and column C
identifies all the area above the Z score. The area in B +
C always sums to 1, the total area of one distribution.
When X = u, Z = 0, column B contains .5000, column C
contains .5000, and the likelihood of being larger that the
u is 50% (as is the likelihood of being smaller).
NOTATION:
To indicate the question, "What is the likelihood that
a score is less than some value?" (ie 10) the notation
would appear:
p(x<10)=_______.
Above, the p stands for probability ( 0r proportion).
The likelihood that a score is greater would appear:
p(x>10)=_______.
Either could be solved by computing the Z-score for X.
This will require a knowledge of u and σ. Once computed,
the z-score could be used to determine the area up to the
desired point. If the Z is negative, the order of columns B
and C are reversed. The Unit Normal Table (UNT) translates
raw scores (X) to likelihoods (p-values) via the Z-scores.
If u = 5 and σ = 5, then 10 is 1 standard of deviation
above the mean , or Z = +1. It is placed at approximately
the 84th%. That is, .8413 or 84.13% of the distribution is
lower than a score of 10. The likelihood of doing better
that 10 is only 15.87%. There is a much greater likelihood
of getting a smaller score than 10 (84.13%, to be exact).
BINOMIALS:
When a question about an outcome can be stated as a
dichotomy (the either/or scenario), the area under the
curve can be approximate from the UNT via the BINOMIAL
formula. The binomial approximation asks a discrete
question of a continuous data set.
If n = the number of trials;
p = the probability of a success;
q = the probability of a failure;
u = n x p
and the sigma of the binomial is the √ of (n)*(p)*(q)
then Z, for the binomial becomes:
Z(Bi) = X - np / σ
Because the binomial approximation asks a discrete
question of a continuous set (the UNT), caution must be
taken to use the upper limit of X when estimating
likelihood above the number, and the lower limit of X is
used when estimating below the number. Further, you can
determine the area under the curve for a discrete number.
That means, in the binomial case, you can determine
p(X<10), p(X>10), p(X=10), p(X<10) and the p(X>10).