Stat: Chapter 4

Basic Properties of Probabilities

1. The probability of an event is always between 0 and 1, inclusive. 2. The probability of an event that cannot occur is 0. (An event that cannot occur is called an impossible event.) 3. The probability of an event that must occur is 1. (An event that must occur is called a certain event.)

probability model

A mathematical description of the experiment based on certain primary aspects and assumptions.

Experiment

An action whose outcome cannot be predicted with certainty.

Independent Events

Event B is said to be independent of event A if P(B|A) = P(B). * One event is independent of another event if knowing whether the latter even occurs does not affect the probability of the former event.

The Complementation Rule

For any event E, P(E) = 1 - P(not E). * The complementation rule is useful because sometimes computing the probability that an event does not occur is easier than computing the probability that is does occur.

The Conditional Probability Rule

If A and B are any two events P(A) > 0, then P(B|A) = P(A&B) / P(A) * The conditional probability of one event given another equals the probability that both events occur divided by the probability of the given event.

The General Addition Rule

If A and B are any two events, then P(A or B) = P(A) + P(B) - P(A & B)

The General Multiplication Rule

If A and B are any two events, then P(A & B) = P(A) * P(B|A).

Probability Notation

If E is an event, then P(E) represents the probability that event E occurs. It is read "the probability of E."

special addition rule

If event A and event B are mutually exclusive then: P(A or B) = P(A) + P(B). More generally, if events A, B, C,... are mutually exclusive, then P(A or B or C or...) = P(A) + P(B) + P(C)

probability

Is a generalization of the concept of percentage. A probability near 0 indicates that the event in question is very unlikely to occur when the experiment is performed, whereas a probability near 1 (100%) suggests that the event is quite likely to occur.

Venn diagrams

Named after English logician John Venn (1834-1923), are one of the best ways to portray events and relationships among events visually. The sample space is depicted as a rectangle, and the various events are drawn as disks (or other geometric shapes) inside the rectangle.

The Equal-Likelihood Model

Probability for Equally Likely Outcomes (f/N Rule): Suppose an experiment has N possible outcomes, all equally likely. An event that can occur is f way has a probability f/N of occurring. * Probability of an event = f/N * f = number of ways event can occur * N=total number of possible outcomes This is an example of a probability model.

Event

Some specified result that may or may not occur when an experiment is performed. A collection of outcomes for the experiment, that is, any subset of the sample space. An event occurs if and only if the outcome of the experiment is a member of the event.

sample space

The collection of all possible outcomes for an experiment. Reflects the fact that, in statistics, the collection of possible outcomes often consists of the possible samples of a given size.

Frequentist interpretation of probability

The probability of an event is the proportion of times the even occurs in a large number of repetitions of the experiment.

Conditional Probability

The probability that event B occurs given that event A occurs is called conditional probability. It is denoted P|B), which is read "the probability of B given A." We call A the given event.

Probability Theory

The science of uncertainty.

What is one of the main goals of statistics?

To be able to use information from a sample, which is part of a population, to draw conclusions about the entire population. (However, we can never be certain that our conclusions are correct since our conclusions are based on only part of the population).

Mutually Exclusive Events

Two or more events are mutually exclusive events if not two of them have outcomes in common.

univariate data

data from one variable of a population

bivariate data

data from two variable of a population

contingency table or two-way table

frequency distribution for bivariate data

joint probabilities

probability of a joint event occurring

marginal probabilities

probability of the events represented in the margins of the contingency table