Probability
Randomness
When we produce data by randomized
procedures, the laws of probability
answer the question,

“What would happen if we did this
many times?
What is probability?
Probability describes only
happens in the long run.
what
Let’s take a look at the probability
applet at www.whfreeman.com/ips
Language of Probability
We call a phenomenon random if
individual outcomes are uncertain but
there is a regular distribution of outcomes
n a large number of repetitions.
The probability of any outcome is the long
term relative frequency.
Probability
Interested in experiments that have more than one
possible outcome.
Examples:
 roll a die
 select an individual at random and measure height
 select a sample of 100 individuals and determine
the number that are HIV positive
We cannot predict the outcome with certainty before
we perform the experiment.
The set of all possible outcomes is called the
sample space, S.
Some experiments consist of a series of
operations. A device called a tree diagram is
useful for determining the sample space.
Any subset of the sample space is called an
event. An event is said to occur if any outcome
in the event occurs.
Two events, A and B, are mutually exclusive,
if they cannot both occur at the same time.
In most experiments the probability function is
unknown.
The probability of an event A, denoted P(A), is
the expected proportion of occurrences of A if the
experiment were performed a large number of
times. The definition implies:

P(S) = 1

P(A or B)=P(A) + P(B) if A and B are mutually exclusive
Compound Events
Event A or B occurs if A occurs, B occurs, or
both A and B occur.
Event A and B occurs if both A and B occur.
Sometimes we wish to know if Event A occurred
given that we know that Event B occurred. The
occurrence of Event A given that we know
Event B occurred is denoted by A|B.
The complement of an Event A , denoted, is all
sample points not in A.
The Addition Rule
The Addition Rule:
P(A or B) = P(A) + P(B) - P(A and B)
If A and B are mutually exclusive, the last term is
zero.
Conditional Probability
The conditional probability of A given B is

P(A|B)=P(A and B)/P(B)

P(B|A)=P(A and B)/P(A)
At times, we can find P(A|B) directly.

Example: Draw two cards without replacement from a
standard deck of cards.
B={1st card is an Ace} and A={2nd card is an Ace}.
P(A|B) = 3/51.
The Complement Rule
The complement Rule:

1 - P(A) = P(A )
Independent Vs. Dependent
Two events are said to be independent if the occurrence
of one does not effect the probability of occurrence of the
other. In symbols, P(A) = P(A|B) and P(B) = P(B|A)
Events that are not independent are called dependent.
Example:
 Draw two cards without replacement
 A and B are dependent.

Suppose we return the 1st card and thoroughly shuffle before
the 2nd draw.
 A and B are independent.
Example
Select an individual at random. Ask place of residence & do you favor
combining city and county government?
Favor,F
Oppose
Total
City,C
80
|
40
| 120
______________|________________|
Outside
|
|
City
20
|
10
| 30
_________________________________________
100 |
50
| 150
P(Favor)=
P(F|C)=P(F and C)/P(C)=
Multiplication Rule:
P(A and B) = P(A) P(B|A) = P(B) P(A|B)
For
independent
events,
P(A and B) = P(A)* P(B)
this
simplifies
Example: Draw two cards without replacement.
A={1st card ace} and
B={2nd card ace}
P(A and B) = P(A)* P(B|A)
 = (4/52)*(3/51) = 12/2652 = .004525
Draw two cards with replacement.
 P(A and B) = P(A)* P(B) = (4/52)*(4/52) = .0059
to