Chapter 3: Probability - 3.1 Random experiments, outcomes And events.

Union

A and B are E's in S. Their union, denoted, A U B, is any E belonging to either A or B. If any event from E1 -> EK, belongs to any E in the union EA -> EK, its with in the union.

Intersection of events

A and B are two (E)vents in the (S)ample space. Their intersection; denoted by an upside down U, is the set of all basic outcomes in S that belongs to both A and B. The probability of intersecting events is called joint probability. Given K events (E1 U E2 U, .... EK), their intersection is is the set of basic outcomes that belongs to every event E1 -> EK.

Complement

A is an E in S. Every set not belonging to A is said to be the A complement or A-bar. A and A-bar are collectively exhaustive and Mutually exclusive.

random experiment

A random experiment is a process leading to two or more possible outcomes, without knowing exactly which outcome will occur. Must lead to a basic outcome.

Collectively Exhaustive

Given the K events E1, E2, . . . , EK in the sample space, S, if E1 U E2 U . . . U EK = S, these K events are said to be collectively exhaustive.

Mutually exclusive sets

If EA and EB do not have any basic outcome in common they are mutually exclusive and their set EA union EB is an empty set containing nothing. The set is only mutually exclusive if each event is mutualy exclusive from E1 ->EK.

The null event

Is denoted as an O with a / in it. Represents the nonoccourence of a basic outcome.

Combinations

N!/x!(n-x)! Implies order is not important and that each item can just be choosen once.

Classic probability

P(A) = NA/N. Probability of A is the number of Events belonging to NA divided by the number of total events.

The three basic probability postulates

S = Sample space Oi= Basic outcomes A = An event For each event A of the sample space, S, we assume that P(A) is defined and we have the following probability rules: 1. If A is any event in the sample space S: 0< p(a)<1 2. . Let A be an event in S and let Oi denote the basic outcomes. Then, P (A) = Sigma (A) P(Oi). 3. P(1) = S

Subjective Probability

Subjective probability expresses an individual's degree of belief about the chance that an event will occur.

Formula for Determining the Number of Combinations

The counting process can be generalized by using the following equation to compute the number of combinations of n items taken x at a time: N!/X!(n-x)! 0! =1 May only be jused once.

Basic outcomes

The possible outcomes in any given random experiment, what could happen. And two or more basic outcomes are not possible.

The relative frequency probability

The relative frequency probability is the limit of the proportion of times that event A occurs in a large number of trials, n.

Sample space

The set of all basic outcomes; denoted with a capital S.

Premutations

The total number of permutations of x objects chosen from n, Pn x, is the number of possible arrangements when x objects are to be selected from a total of n and arranged in order. This is granted that we can only be chosen once. N!/(n-x)!

Number of orderings

x! X factorial.

Event

Denoted E. is any subset of the basic outcomes FROM the sample space.