A (Brief) General Discussion of
Probability: Some “Probability Rules”
Some abstract math language too! (from various internet sources)


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.
2

•
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.
3

Example: Flipping a Fair Coin

• 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

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!

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.

• 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