Probability

1
Probability
2
Randomness

• When we produce data by randomized procedures,
  the laws of probability answer the question,
• What would happen if we did this many times?

3
What is probability?

• Probability describes only what happens in the
  long run.
• Lets take a look at the probability applet at
  www.whfreeman.com/ips

4
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.

5
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.

6

• 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.

7

• 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.

8

• 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

9
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 AB.
•  
• The complement of an Event , denoted, is all
  sample points not in A.

10
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.

11
Conditional Probability

• The conditional probability of A given B is
• P(AB)P(A and B)/P(B)
• P(BA)P(A and B)/P(A)
•  
• At times, we can find P(AB) directly.
• Example Draw two cards without replacement from
  a standard deck of cards.
• B1st card is an Ace and A2nd card is an
  Ace.
• P(AB) 3/51. 

12
The Complement Rule

• The complement Rule
• 1 - P(A) P( )

13
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(AB) and P(B) P(BA)
•  
• 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.
•  

14
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(FC)P(F and C)/P(C) 

15
Multiplication RuleP(A and B) P(A) P(BA)
P(B) P(AB)

• For independent events, this simplifies to
  P(A and B) P(A) P(B)
•  
• Example Draw two cards without replacement.
• A1st card ace and
• B2nd card ace
•  
• P(A and B) P(A) P(BA)
• (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