Basic Ideas in Probability Study Guide

Updated on Oct 5, 2011

Introduction to Basic Ideas in Probability

For most studies, we want to use the study to make decisions or to draw inferences about the research question. Before we are prepared to do this, we need to establish some understanding of probability, discrete and continuous distributions, and sampling distributions. These will be the topics of this lesson. We will begin by first introducing some probabilistic concepts.

Sample Spaces and Events

People use probability to make decisions daily. The weatherman says that the chance of rain is 60% in the afternoon, and you decide it may be a good idea to take an umbrella. An economist predicts that the stock market will go up during the next six months, and you decide to buy some stocks. A friend took a class last semester that you are now taking.He says that he is sure you can pass without studying for any test, and you must decide whether or not to study. Some food in the refrigerator is past its expiration date, but it smells okay. Do you eat it or not? Based on our past experiences, we make these decisions, primarily without thinking about probability. Now, we will begin to look at probability more formally.

To begin, suppose we roll a fair die and observe the number of dots on the upper face. This is a random experiment because a chance process (rolling the die) determines the outcome. The set of all possible outcomes is called the sample space and will be denoted by S. For our die rolling experiment, the sample space is S1 = {1, 2, 3, 4, 5, 6}.

Suppose that we collect fish randomly from a lake and measure their lengths to the nearest inch. The sample space for this random experiment may be S2 = {1, 2, 3, ... }. On the other hand, a more appropriate sample space could be S3 = {x|x > 0},which is read as "the set of all x such that x > 0." This sample space reflects the fact that length is not always, in fact is seldom, an integer.Whether S2 or S3 should be used in a particular problem depends on the nature of the measurement process. If decimals are to be used, we need S3. If only integers are to be used, S2 will suffice. The point is that sample spaces for a particular experiment are not unique and must be selected to provide all pertinent information for a given situation.

Consider the die rolling experiment again. Suppose we are interested in obtaining a 1 on a single roll; that is, rolling a 1 is an event of interest. Other possible events are rolling an even number, rolling a number less than 4, and so on. Formally, an event is any subset of the sample space. Because the null set (the set with no outcomes and often denoted by ø) is a subset of every set, the null set is an event associated with every sample space. Similarly, because every set is a subset of itself, the sample space is an event in every experiment. If the outcome of an experiment is contained in the event E, then the event E is said to have occurred. Events are usually denoted with capital letters at the beginning of the alphabet.

Returning to the die rolling experiment, let A be the event of rolling a 1; that is, A={1}. Define B={2, 4, 6}, the event of obtaining an even number on the roll. And define C = {1, 2, 3}, the event of rolling a number less than 4. If the die is rolled, and a 2 appears on the upper face, then events B and C have both occurred. Note: It is enough that one of the outcomes in the event occurs for the event to occur; not all elements of the event must be observed (in fact, they cannot).

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