The mathematics of probability is expressed most naturally in terms of sets. This chapter lays out the basic terminology and reviews naive set theory.
A set is a collection of things, without regard to their order. There are several simple operations on sets that yield other sets, and there are names for special relationships between sets. This section reviews the definitions and operations, and how to manipulate them. These definitions and operations are part of the branch of Mathematics called Set Theory. Translating word problems into the language of set theory is crucial in solving probability problems. Venn diagrams provide a way to visualize sets and relationships among sets. The following table gives some of the definitions used in Set Theory.
Venn diagrams represent sets and the relationships among sets pictorially. A Venn diagram represents the universe S by a two-dimensional region (usually a rectangle or a circle). Subsets of S are represented by sub-regions. The complement of the set A is represented by everything in the region that represents S that is not in the sub-region that represents A. The overlap of the sub-region that represents the set A with the sub-region that represents the set B represents the intersection AB, and so on. Shading or highlighting is used in Venn diagrams to draw attention to special relationships or subsets. contains a Venn diagram that displays the universe S and two subsets of S, A and B. The figure has check boxes on the right side. Checking a box highlights the corresponding set; the options are as follows:
You can click and drag A and B to change the amount by which they intersect. Scrollbars at the bottom of the figure let you change the sizes of A and B. Usually, Venn diagrams do not have a scale, but because this figure is intended to represent the outcome space S, it represents the probability of an event by the area of the event. Probability is never greater than 100%, so the area of S in the diagram is 100%. The area of the subsets of S represents their probability, so the areas of A and of B are between 0% and 100%; these are denoted P(A) and P(B) at the bottom of the figure. The areas of the events AB and A∪B are listed in the figure as P(AB) and P(A or B), respectively.
You need Java to see this.
A random experiment or random trial is basically any situation whose outcome is not perfectly predictable, but for which we can specify all possible outcomes, and that shows long-term regularities. For example, when we toss a coin, we do not know how it will land, but it certainly must land heads, tails, on its edge, or not land at all. There is no other possibility. The set of all possible outcomes of a random experiment is called the outcome space. The letter S will denote outcome space. We are free to choose the outcome space to correspond to what we deem relevant for the experiment, as long as it is essentially inevitable that the random experiment will result in some outcome in the outcome space. For example, the outcome space we just described was {heads, tails, edge, doesn't land}. It might be adequate for our purposes for the outcome space to be {heads, not heads}.
Often we shall tailor outcome spaces for specific problems. Here is an example: Imagine a box containing tickets that are indistinguishable except that each has written upon it a unique number between 1 and the number of tickets, n. We shake the box, draw a ticket from the box without looking, and record the number written on the ticket we happened to draw. The natural outcome space of this experiment is the set of numbers {1, 2, … , n}. However, suppose we are interested only in whether the number on the ticket we draw is even. The outcome space then could be reduced to {even number on ticket, odd number on ticket}, or coded even more abstractly as {0, 1}, where the outcome is the number of even-numbered tickets drawn.
An event is a subset of outcome space: a collection of outcomes in the outcome space. A is an event if A⊂S. For example, in the experiment of drawing a numbered ticket from the box, suppose there are 10 tickets in all, and that we choose the outcome space S to be the numbers {1, 2, 3, … , 9, 10}. Then "we draw the number 1" is the event {1}, and "we draw an even number" is the event {2, 4, 6, 8, 10}, both of which are subsets of the set of possible outcomes, the outcome space S.
Two events are disjoint or mutually exclusive if the occurrence of one is incompatible with the occurrence of the other; that is, if they have no outcome in common. This is equivalent to the definition of disjoint sets, viewing events as sets. The event A implies the event B if A⊂B: then if A occurs, B must also occur (if the outcome that occurs is in A, the outcome that occurs is also in B, because every element of A is an element of B).
The following exercises check your ability to translate word problems into the language of set theory.
Sets are collections of things without regard to their order. Sets can be specified by listing their elements or by giving a property satisfied by their elements (and nothing else). The empty set, the set with no members, is {}. Set operations turn sets into other sets. If x is in the set A we write x∈A or A∋x. The complement of a set A, Ac, relative to a "universe" of elements S, is the set of elements of the reference universe S that are not in A. The set A is a subset of the set B (A⊂B) if every element of A is also an element of B. The intersection of the set A and the set B, AB, is the set of things that are in both A and B. The union of the set A and the set B, A∪B, contains every element of A, every element of B, and nothing else. A and B are disjoint or mutually exclusive if they have no elements in common, that is, if AB = {}. The collection {A1, A2, A3, … } exhausts A if every element of A is in at least one of the sets in the collection; that is, if A = A1 ∪ A2 ∪ A3∪ … . The collection {A1, A2, A3, … } is disjoint if AiAj = {} when i ≠ j. The collection {A1, A2, A3, … } partitions A if every element of A is in exactly one of the sets in the collection; that is, if the collection is disjoint and exhausts A.
Outcome space S is the set of all things that could possibly happen in an experiment. Events are subsets of S. The event A occurs if the outcome of the experiment is an element of A; otherwise, A does not occur. Two events are disjoint or mutually exclusive if they have no outcomes in common—if they are disjoint subsets of S. Disjoint events cannot occur simultaneously. For example, if a coin is tossed, the event that the coin lands "heads" and the event that the coin lands "tails" are disjoint.
Set theory forms the basis of the mathematical study of probability, which we begin in Chapter 11, "Probability: Axioms and Fundaments."