Probability, An Introduction
Probability is as intimately related to counting processes as you are to your skeleton!
Without counting we cannot cover the concept except in fuzzy generalities. By the end
of this section you should believe this.
However, we do need to recognize some differences between our counting vocabulary and the probability
vocabulary. Let’s begin with a definition of probability.
Probability is the study of the likelihood or chance that an outcome or event can occur.
This sentence is replete with words that need explanation!
An
outcome
is some basic result that we can record in a process.
Example:
Say we flip a coin. The
outcomes
are to see
heads
or
tails
. We discount the
on-edge
possibility as a
nonevent.
Example:
We spin a dial with areas marked
a, b, c
. Then the outcomes are
a, b,
or
c
.
A
sample space
,
usually denoted by
S,
is the collection of all possible
outcomes
from some action or process.
Example:
Say we flip a coin. The outcomes are a
heads
or a
tails
. These are the only two possible outcomes.
We discount “edgies” and require a re-toss just as in most sporting events.
S = {heads, tails}
Example:
Say we flip a coin two times. The outcomes are of each spin are
heads
or
tails
. However, the sample
space is composed of the results of both spins. We list them as ordered pairs.
S = {(heads, heads), (heads, tails), (tails, heads) (tails, tails)}
The number of elements in the sample space is simply an application of previous counting methods. There
are two choices at each flip. So there are
outcomes.
2
22 2 4
Example:
We spin a dial with areas marked
a, b, c
.
S =
{a, b, c}
.
Example:
We spin a dial with areas marked
a, b, c
three times. Similarly to the coin flip the outcomes are not
a, b,
or
c
. They are the ordered triples listed as (
first spin, second spin, third spin
). The sample space
is listed in the table below.
aaa
aab
aac
aba
abb
abc
aca
acb
acc
baa
bab
bac
bba
bbb
bbc
bca
bcb
bcb
caa
cab
cab
cba
cbb
cbc
cca
ccb
ccc
Listing the 27 outcomes is tedious. Fortunately, we are more interested than the
count
of the set than the list
itself in many cases. Again counting the sample space is an application of the multiplication principle.
There are three choices at each spin. The sample space has
outcomes.
3
3333 2
7