Notes
1 1 Source: Reproduced with permission of ASQ, Jacobson (1998).
2 2 Source: Based on data from The Engineering Statistics Handbook, National Institute of Standards and Technology (NIST).
Chapter 3 Elements of Probability
The focus of this chapter is the study of basic concepts of probability.
Topics Covered
Random experiments and sample spaces
Representations of sample spaces and events using Venn diagrams
Basic concepts of probability
Additive and multiplicative rules of probability
Techniques of counting sample points: permutations, combinations, and tree diagrams
Conditional probability and Bayes's theorem
Introducing random variables
Learning Outcomes
After studying this chapter, the reader will be able to
Handle basic questions about probability using the definitions and appropriate counting techniques.
Understand various characteristics and rules of probability.
Determine probability of events and identify them as independent or dependent.
Calculate conditional probabilities and apply Bayes's theorem for appropriate experiments.
Understand the concept of random variable defined over a sample space.
3.1 Introduction
In day‐to‐day activities and decisions, we often confront two scenarios: one where we are certain about the outcome of our action and the other where we are uncertain or at a loss. For example, in making a decision about outcomes, an engineer knows that a computer motherboard requires four RAM chips and plans to manufacture 100 motherboards. On the one hand, the engineer is certain that he will need 400 RAM chips. On the other hand, the manufacturing process of the RAM chips produces both nondefective and defective chips. Thus, the engineer has to focus on how many defective chips could be produced at the end of a given shift and so she is dealing with uncertainty.
Probability is a measure of chance. Chance, in this context, means there is a possibility that some sort of event will occur or will not occur. For example, the manager needs to determine the probability that the manufacturing process of RAM chips will produce 10 defective chips in a given shift. In other words, one would like to measure the chance that in reality, the manufacturing process of RAM chips does produce 10 defective chips in a given shift. This small example shows that the theory of probability plays a fundamental role in dealing with problems where there is any kind of uncertainty.
3.2 Random Experiments, Sample Spaces, and Events
3.2.1 Random Experiments and Sample Spaces
Inherent in any situation where the theory of probability is applicable is the notion of performing a repetitive operation, that is, performing a trial or experiment that is capable of being repeated over and over “under essentially the same conditions.” A few examples of quite familiar repetitive operations are rolling a die, tossing two coins, drawing five screws “at random” from a box of 100 screws, dealing 13 cards from a thoroughly shuffled deck of playing cards, filling a 12‐oz can with beer by an automatic filling machine, drawing a piece of steel rod, and testing it on a machine until it breaks, firing a rifle at a target 100 yards away, and burning ten 60‐W bulbs with filament of type
An important feature of a repetitive operation is illustrated by the repetitive operation of firing a rifle at a 100‐yard target. The shooter either hits the target or misses the target. The possible outcomes “hit” or “miss” are referred to as outcomes of the experiment “firing at a target 100 yards away.” This experiment is sometimes called a random experiment. We will have more discussion of this at a later point. This important feature needs formalizing with the following definition.
Definition 3.2.1 In probability theory, performing a repetitive operation that results in one of the possible outcomes is said to be performing a random experiment.
One of the basic features of repetitive operations or random experiments under specified conditions is that an outcome may vary from trial to trial. This variation leads to the analysis of the possible outcomes that would arise if a trial were performed only once. The set of all possible outcomes under specific conditions if an experiment was performed once is called the sample space of the experiment and is denoted by S. It is convenient to label an outcome in a sample space S by the letter e, and call e a sample space element or simply an element or sample point of the sample space S. The sample space S of such elements or points is generated by the operations or trials of a random experiment. Consider the following examples of elements or sample points that constitute a sample space.
Example 3.2.1 (Rolling a die) If a die is rolled once, the sample space S thus generated consists of six possible outcomes; that is, the die can turn up faces numbered 1, 2, 3, 4, 5, or 6. Thus, in this case,
Example 3.2.2 (Tossing two coins) If two coins are tossed, say a nickel and a dime, and if we designate head and tail on a nickel by H and T, respectively, and head and tail on a dime by h and t, respectively, the sample space S generated by tossing the two coins consists of four possible outcomes. We then have that
As an example, Ht denotes the outcome that the nickel, when tossed ended up showing head, while the dime, when tossed, showed tail.
Example 3.2.3 (Sample space for item drawn using random sampling scheme) The sample space
Example 3.2.4 (Sample space for playing cards) In dealing 13 cards from a thoroughly shuffled deck of ordinary playing cards, the sample space