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 what happens in the long run. 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 , 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( )

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 Favor,F Oppose Total City,C 80 | 40 | 120 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, this simplifies to P(A and B) = P(A)* P(B)   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