3.1 The
Concept of Probability

Probability
deals with the chance that something will happen. It is always a
number between 0 and 1 with the greater the chance that it will
happen the closer the number will get to 1. In statistics we will
sometimes study this by doing changes in a controlled environment.
This would be performing a controlled experiment. We can also
observe uncontrolled environments, for example a sports team or stock
prices. In doing so we are doing an experiment.

An experiment
is any process of observation that has an uncertain outcome. The
process must be defined so that on any single repetition of the
experiment, one and only one of the possible outcomes will
occur. The possible outcomes for an experiment are called
experimental outcomes.

Our assigning a value between 0 and 1 as noted above, is assigning
the probability that a thing will happen. All probabilities must add
up to 1.

When the possibilities of an event happening are equally likely, we
use the classic method to assign probabilities of events happening.
For example, a coin toss should have a 1 in 2 chance of being a head
and the same for tails. This is just for one run of the event.

We often look at long run relative frequencies. If we got 6 heads in
a coin toss, then our probability would be .6. The more the event is
repeated, the closer it should come to the estimate that we create
from classic method. In reality you cannot repeat an event enough to
make this happen.

Sometimes the classic method can not be used to determine an outcome,
in those cases we will need to use a method to see what the results
would be. For example, select a random group of consumers and ask if
they like product A or B better. If they like A 17 out of 100
people, then it would have .17. These would be made into a relative
frequency. It may be necessary to estimate this probability since we
may not be able to produce enough experimental attempts.

3.2 Sample Space and Events

The
sample
space of an
experiment is the set of all possible experimental outcomes. The
experimental outcomes in the sample space are often called sample
space outcomes.

To figure these things out, we can assign possibilities based on
random events like we did earlier. If we toss a coin twice the
possibility that we will have exactly 2 heads is 1 in 4. This can be
proved by creating a tree: It can either be a head or a tail. Each
of these can only produce a head or a tail. That makes 2
possibilities for each of the two possibilities or a total of 4
possible outcomes.

An
event
is a set (or collection) of sample space outcomes.

The
probability
of an event.
is the sum of the probabilities of the sample space
outcomes that correspond to the
event.

If A is an
event, then 0 ≤ P(A) ≤ 1.

Moreover:

1. 
   If an event never occurs, then the probability of this event equals
   0.
   

   
   

2. If an event is
   certain to occur, then the probability of this event equals 1.

   
   

If all of the
sample space outcomes are equally likely, then the probability
that an event will occur is equal to the ratio

3.3 Some
Elementary Probability Rules

We use several
rules in determining probability.

Te simplest rule is
that given an event A, the compliment of, or the not occurring events
are called A-bar. So P(A) is the chance the event will occur and
P(Abar) is the chance that it will not. Further more P(A) and
P(Abar) when added will equal 1.

The Rule of
Compliments:

P(Abar)
= 1 – P(A)

Using
our coin toss, if the event of two heads is .25 (1 in 4 or 25%, then
the compliment must be:

P(Toss-bar)
= 1 – p(Toss)

= 1 –
p(.25)

= .75

Another
rule: to understand know that U is all events that are in A or B and
that ^ is all the events that are in A and B

The
Intersection and Union of Two Events

Given two events A
and B,

l   The
intersection of A and B is the event consisting of the sample
space outcomes belonging to both A and B. The
intersection is denoted by A ^ B . Furthermore, P(A^B)
denotes the probability that both A and B will simultaneously
occur.

l   The
union of A and B is the event consisting of the
sample space outcomes belonging to A or B (or both).
The union is denoted A U B. Furthermore, P(A U B) denotes the
probability that A or B (or both) will occur.

The
Addition Rule

Let A and B
be events. Then, the probability that A or B (or both) will
occur is

P(A U B) = P(A) +
P(B) – P(A ^ B)

This is true
because in adding A and B we add where the intersect as well,
therefore we must subtract it back out. If events are mutually
exclusive (it has to be one or the other) than P(A ^ B) = 0. This
would mean P(A ^ B) = P(A) + P(B).

The
Addition Rule for N Mutually Exclusive Events

The events A₁,
A₂, ...,
A_N are
mutually exclusive if no two of the events have any sample space
outcomes in common. In this case, no two of the events can occur
simultaneously, and

P(A1 U A2 U ... U
An) = P(A1) + P(A2) + ... + P(An)

3.4 Conditional
Probability and Independence

Conditional
probability deals with if an even can happen if another event has
already happened. For example, if I draw a king out of a deck of
cards, what is the probability that I will draw another king. It is
written P(A | B) spoken 'probability of A given B”.

Conditional
Probability

1. The
   conditional probability that A will occur given that B
   will occur is written P (A | B)
   and is defined to be
   

   
   

P (A | B) = P(A ^ B)/P(B)

Here we assume that P(B) is greater than 0.

2. The
   conditional probability that B will occur given that A
   will occur is written P(B | A)
   and is defined to be

   
   

P(B |
A) P(A ^ B)/P(A)

Here we assume that
P(A) is greater than

The
General Multiplication Rule—Two Ways to Calculate P (A
∩ B)

Given any two
events A and B,

P(A ^
B) = P(A)P(B | A)

=
P(B)p(A | B)

Independent
Events

Two events A
and B are independent if and only if

1. 
   P(A | B) =P(A) or, equivalently,
   

   
   

2. P(B
   | A) =P(B)

   
   

Here we
assume that P(A) and P(B) are greater
than 0.

The
Multiplication Rule for Two Independent Events

If A and B
are independent events, then

P(A ^
B) = P(A)P(B)

Here we
could use an example. If the chance of the cafateria serving sloppy
joes is .6 and the chance of the statistics teacher giving a pop quiz
are .7 then since these should be independent of each other P(C ^ P)
= P(.6) P(.7) = .42

If we
have multiple independent events we just multiply the numbers out.