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A (Brief) General Discussion of
Probability: Some “Probability Rules”
Some abstract math language too! (from various internet sources)
Probability 
The Science of Random Behavior
• Random behavior is unpredictable for an
individual object, but it has a regular and
predictable pattern on the average for a
huge number of objects.
• This is why we can use probability to gain useful
results from random samples & randomized
comparative experiments.
• Random  Individual outcomes are uncertain
but there is a regular distribution of outcomes in a
large number of repetitions.
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• Relative Frequency  The proportion of
occurrences of an outcome. It settles down
to one value over the long run.
That one value is then defined to be
the probability of that outcome.
• Relative Frequency Probabilities can be
determined (checked) by observing a long
series of independent trials (empirical data):
– Do experiments with many samples
– Do simulations, with computers, with random
number tables, etc.
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Example: Flipping a Fair Coin
Probability Models
• The sample space S of a random
phenomenon is the set of all possible
outcomes.
• An event is an outcome or a set of outcomes
(a subset of the sample space).
• A probability model is a mathematical
description of long-run regularity consisting
of a sample space S and a method of
assigning probabilities to events.
Probability Model for Two Fair Dice
Example of Random Phenomenon: Roll a pair of fair dice.
The Sample Space is illustrated in the figure:
The probabilities of each individual of the 36 outcomes are
found by inspection. Each clearly occurs with a probability:
p = (1/36) = 0.0278
Probability Rule #1:
All Probabilities Must Be Numbers
Between 0 & 1.
• A probability can be interpreted as the
proportion of times that a certain event can be
expected to occur.
• So, if the probability of an event is more than
1, then it will occur
more than 100% of the time!! This is
clearly illogical & impossible!
Probability Rule #2:
The sum of the probabilities of all
possible outcomes must be 1.
• Because some outcome must occur on every
trial, the sum of the probabilities for all
possible outcomes must be exactly one.
• If the sum of all of the probabilities is less
than one or greater than one, then the
resulting probability model will be
incoherent & illogical.
Probability Rule #3:
• If two events have no outcomes in
common, they are said to be
disjoint.
• The probability that one or the
other of two disjoint events occurs
is the sum of their individual
probabilities.
Example of Disjoint Probabilities:
• If two events have no outcomes in common, they are disjoint.
Probability that one or the other of 2 disjoint events
occurs = sum of individual probabilities.
• Consider data about the age distribution of women at first child
birth. Obviously, the sample population is women have had
children! So, it excludes women with no children!
• A study of census data gives:
Under 20: 25%. 20-24: 33%. 25+: ?
• So, the probability that a woman
in the sample population who is 24
or younger has had a first child is:
= 25% + 33% = 58%
Note: By Rule #3
(or Rule #2)
This must be 42%
Probability Rule #4:
The probability that an event DOES NOT
OCCUR  1 minus the probability that
the event DOES OCCUR.
• Example: As a jury member, you assess the
probability that the defendant is guilty to be 0.8.
So, you must also believe that the probability the
defendant is not guilty is 0.2 in order to be
consistent.
• Similarly, if the probability that a flight will be
on time is 0.7, then the probability it will be late
is 0.3.
Summary:
Probability Rules in Mathematical Notation
Consider again the Random Phenomenon of rolling a
pair of fair dice. As we’ve already seen,
The Sample Space is illustrated in the figure:
The probabilities of each individual of the 36 outcomes are
found by inspection. Each clearly occurs with a probability:
p = (1/36) = 0.0278
All possible outcomes of rolling a pair of fair dice
are shown in the figure.
Calculate the probability P(5) of rolling a 5.
All possible outcomes of rolling a pair of fair dice
are shown in the figure.
Calculate the probability P(5) of rolling a 5.
P(5)  P(
) + P(
) + P( ) + P( ) =
All possible outcomes of rolling a pair of fair dice
are shown in the figure.
Calculate the probability P(5) of rolling a 5.
P(5)  P(
) + P(
) + P( ) + P( ) =
(1/36) + (1/36) + (1/36) + (1/36) = (4/36) =
= 0.111