Probability You’ll probably like it!

Probability Definitions Probability assignment Complement, union, intersection of events Conditional probability and independence Combinations

Probability - Some Definitions An experiment is the process of observing a phenomenon with multiple possible outcomes The sample space of an experiment is all possible outcomes –The sample space may be discrete or continuous An event is a set (collection) of one or more outcomes in the sample space

The probability of an event is the proportion of times the event is expected to occur in repeated experiments

Probability Properties The probability of an event, say event A, is denoted P(A). All probabilities are between 0 and 1. (i.e. 0 < P(A) < 1) The sum of the probabilities of all possible outcomes must be 1.

Assigning Probabilities Guess based on prior knowledge alone Guess based on knowledge of probability distribution (to be discussed later) Assume equally likely outcomes Use relative frequencies

Guess based on prior knowledge alone Event B = {It rains Tomorrow} Weth R. Guy says “There is a 30% chance of rain tomorrow.” P(B) =.30

Assume equally likely outcomes

Use Relative Frequencies Shut this thing off and make up examples on the blackboard.

Complement* The complement of an event A, denoted by A, is the set of outcomes that are not in A A means A does not occur * Some texts use A c to denote the complement of A

Law of Complement P(A) = 1 - P(A)

Union The union of two events A and B, denoted by A U B, is the set of outcomes that are in A, or B, or both If A U B occurs, then either A or B or both occur

Intersection The intersection of two events A and B, denoted by AB, is the set of outcomes that are in both A and B. If AB occurs, then both A and B occur

Addition Law P(A U B) = P(A) + P(B) - P(AB) (The probability of the union of A and B is the probability of A plus the probability of B minus the probability of the intersection of A and B)

Mutually Exclusive Events* Two events are mutually exclusive if their intersection is empty. Two events, A and B, are mutually exclusive if and only if P(AB) = 0 * The book also uses the term incompatible events

Addition Law for Mutually Exclusive Events P(A U B) = P(A) + P(B)

Conditional Probability The probability of event A occurring, given that event B has occurred, is called the conditional probability of event A given event B, denoted P(A|B)

Conditional Probability P(AB) P(A|B) = P(B) or P(AB) = P(B)P(A|B)

Independence Two events A and B are independent if P(A|B) = P(A) or P(B|A) = P(B) or P(AB) = P(A)P(B)

If P(A|B) = P(A) or P(B|A) = P(B) or P(AB) = P(A)P(B) then A and B are independent.