Ch 2: Probability

unconditional probability

probability based on the entire sample space

The Axioms of Probability

1. Let S be a sample space. Then P(S) = 1. 2. For any event A,0≤ P(A)≤1. 3. If A and B are mutually exclusive events, then P(A ∪ B) = P(A) + P(B). More generally, if A1, A2, . . . are mutually exclusive events, then P(A1 ∪ A2 ∪ ···) = P(A1) + P(A2) + ··· .

mutually exclusive

-The events A and B are said to be mutually exclusive if they have no outcomes in common. -More generally, a collection of events A1, A2, . . . , An is said to be mutually exclusive if no two of them have any outcomes in common.

conditional probability

A probability that is based on a part of a sample space is called a conditional probability

random variable

A random variable assigns a numerical value to each outcome in a sample space.

The Fundamental Principle of Counting

Assume that k operations are to be performed. If there are n1 ways to perform the first operation, and if for each of these ways there are n2 ways to perform the second operation, and if for each choice of ways to perform the first two operations there are n3 ways to perform the third operation, and so on, then the total number of ways to perform the sequence of k operations is n1n2 · · · nk .

combination

Each distinct group of objects that can be selected, without regard to order, is called a combination.

cumulative distribution function

Function which specifies the probability that a random variable is less than or equal to a given value. The cumulative distribution function of the random variable X is the function F(x) = P(X ≤ x).

independent

In this case the conditional and unconditional probabilities are the same, and the events are said to be independent

complement

The complement of an event A, denoted Ac, is the set of outcomes that do not belong to A. In words, Ac means "not A." Thus the event Ac occurs whenever A does not occur.

intersection

The intersection of two events A and B, denoted A ∩ B, is the set of outcomes that belong both to A and to B. In words, A ∩ B means "A and B." Thus the event A ∩ B occurs whenever both A and B occur.

continuous random variable

The possible values of a continuous random variable always contain an interval, that is, all the points between some two numbers. A continuous random variable is defined to be a random variable whose probabilities are represented by areas under a curve

sample space

The set of all possible outcomes of an experiment is called the sample space for the experiment.

union

The union of two events A and B, denoted A ∪ B, is the set of outcomes that belong either to A, to B, or to both. In words, A ∪ B means "A or B." Thus the event A ∪ B occurs whenever either A or B (or both) occurs.

experiment

a process that results in an outcome that cannot be predicted in advance with certainty

probability density function (probability distribution)

the curve whose areas under which are the probabilities of a continuous random variable

probability

the probability is a quantitative measure of how likely the event is to occur

exhaustive

their union covers the whole sample space

discrete random variable

A random variable is discrete if its possible values form a discrete set. This means that if the possible values are arranged in order, there is a gap between each value and the next one. The set of possible values may be infinite; for example, the set of all integers and the set of all positive integers are both discrete sets.

event

A subset of a sample space is called an event.

probability mass function

The list of possible values 0, 1, 2, 3, along with the probabilities for each, provide a complete description of the population from which X is drawn. (see probability distribution)

permutation

an ordering of a collection of objects