Created by Dane McGuckian
Statistics Exam Two Notes
Events, Sample Spaces and Probability
This section introduces the basic concept of the probability of an event.
Three different
methods for finding probability values will be presented.
The most important objective of this section is to learn how to interpret probability
values.
The
Rare Event Rule
for Inferential Statistics
If under a given assumption, the probability of a particular observed event is extremely
small, we conclude that the assumption is probably not correct.
An example of the Rare Event Rule would be as follows:
Say that you assume that a
college graduate will have a starting salary of 75k or more, but a random survey of 32
recent graduates indicates that the starting salaries were around 35k.
If your assumption
is actually true the probability that a sample of 32 recent grads would have an average
salary of only 35k would be extremely small, so we must conclude your assumption was
wrong (in actuality, we need to know what the standard deviation is before we could
decide how probable the above sample results would be, but we will get to that later).
Before we get to probability, there are some terms we need to discuss:
An
experiment
is an act of observation that leads to a single outcome that cannot be
predicted with certainty.
An
event
is a specific collection of sample points.
For example: Event
A
: Observe an even number.
A
sample point (or simple event)
is the most basic outcome of an experiment.
For
example, getting a four on a single roll of a die.
Sample point ~ outcome
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A
sample space
(denoted S) is the collection of all possible outcomes of an experiment.
For example: a roll of a single die:
S: {1, 2, 3, 4, 5, 6}
Example 28: List the different possible families that can occur when a couple has three
children…
Example 29: List the possible outcomes for three flips of a fair coin…
Solution:
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
Here are some commonly used notational conventions:
P
 denotes a probability.
A, B
, and
C
 denote specific events.
P
(
A
)  denotes the probability of event
A
occurring.
The
probability of an event A
is calculated by summing the probabilities of the sample
points in the sample space for A.
In other words:
( )
n
P A
N
=
, n = # of times you observed event A (number of ways A can happen), N =
number of observations (Number of total possibilities).
You might have noticed that the statements in the parenthesis in the above definition
seem to define a second definition of probability. That is because there are two ways to
think about probability.
I have those two ways defined below:
Rule 1: Relative Frequency Approximation of Probability
Conduct (or observe) a procedure, and count the number of times event
A
actually occurs.
Based on these actual results,
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 Winter '08
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 Statistics, Probability, Probability theory, Discrete probability distribution, Dane McGuckian

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