Chapter 5 Summary

The language of probability In Chapter 5, we began by defining a random phenomenon as an event with an outcome that can’t be predicted ahead of time. The long-term behavior of the random phenomenon is described by a number called its probability. A probability experiment is the process of observing the outcome of a random phenomenon. An outcome is any possible result of the experiment. An event is a collection of one or more outcomes in an experiment.

The three basic rules The three most basic probability rules are the following:
(1) The probability of each event is a number between 0 and 1, inclusive;
(2) The sum of the probabilities for all possible outcomes of the event is 1; and
(3) The probability that an event does not occur is 1 minus the probability that it does occur.

Three kinds of probability Probabilities of events can be found using theoretical, empirical, or subjective methods. Theoretical probabilities are derived using the rules of probability. Empirical probabilities are arrived at through experimentation and calculating long-term relative frequencies. Subjective probability occurs when theoretical and empirical methods cannot be use; they are based on a person’s judgment about how likely it is that the event will occur.

Observing many, many trials of an experiment The Law of Large Numbers states that as we observe a random event over more and more trials, the percentage of time that a particular outcome occurs gets closer and closer to the percentage that we expect.

A sample space and its elements The set of all possible outcomes that can result from a probability experiment is called the sample space. A helpful (and systematic) way to identify the possible outcomes from a sample space is to use a tree diagram. A tree diagram displays “branches,” each of which represent a possible outcome for a trial of the experiment. If the outcomes are each equally likely to occur, then the probability that event E is P(E) = (number of successful outcomes)/(total number of outcomes).

Independent and disjoint events Two events are called independent if the outcome of the first has no effect on the outcome of the second. Provided that events E and F are independent, P(E and F) = P(E) · P(F). The probability that either event or event F occurs is given by the following rule: P(E or F) = P(E) + P(F) - P(E and F). Two events are called disjoint if P(E and F) = 0.

Contingency and conditional probability A contingency table (also known as a two-way table) is a classification of the individuals in a sample or a population according to two categorical variables. Empirical probabilities can be calculated using the data in the contingency table. We can also use these data to calculate the conditional probability that event E occurs given that event F (denoted P(E|F). In general, P(E|F) ≠ P(F|E).