5.1: Events

Like many branches of mathematics or science, probability theory has its own nomenclature in which a word may mean something different from or more specific than its everyday meaning. Consider the two words event, which has several everyday meanings, and outcome. Merriam-Webster’s Collegiate Dictionary gives these definitions that are closest to the technical meaning in probability theory:

• outcome: something that follows as a result or consequence
• event: a subset of the possible outcomes of an experiment

In our context, outcome is the symbol selected, whether or not it is known to us. While it is wrong to speak of the outcome of a selection that has not yet been made, it is all right to speak of the set of possible outcomes of selections that are contemplated. In our case this is the set of all symbols. As for the term event, its most common everyday meaning, which we do not want, is something that happens. Our meaning, which is quoted above, is listed last in the dictionary. We will use the word in this restricted way because we need a way to estimate or characterize our knowledge of various properties of the symbols. These properties are things that either do or do not apply to each symbol, and a convenient way to think of them is to consider the set of all symbols being divided into two subsets, one with that property and one without. When a selection is made, then, there are several events. One is the outcome itself. This is called a fundamental event. Others are the selection of a symbol with particular properties.

Even though, strictly speaking, an event is a set of possible outcomes, it is common in probability theory to call the experiments that produce those outcomes events. Thus we will sometimes refer to a selection as an event.

For example, suppose an MIT freshman is selected. The specific person chosen is the outcome. The fundamental event would be that person, or the selection of that person. Another event would be the selection of a woman (or a man). Another event might be the selection of someone from California, or someone older than 18, or someone taller than six feet. More complicated events could be considered, such as a woman from Texas, or a man from Michigan with particular SAT scores.

The special event in which any symbol at all is selected, is certain to happen. We will call this event the universal event, after the name for the corresponding concept in set theory. The special “event” in which no symbol is selected is called the null event. The null event cannot happen because an outcome is only defined after a selection is made.

Different events may or may not overlap, in the sense that two or more could happen with the same outcome. A set of events which do not overlap is said to be mutually exclusive. For example, the two events that the freshman chosen is (1) from Ohio, or (2) from California, are mutually exclusive.

Several events may have the property that at least one of them happens when any symbol is selected. A set of events, one of which is sure to happen, is known as exhaustive. For example, the events that the freshman chosen is (1) younger than 25, or (2) older than 17, are exhaustive, but not mutually exclusive.

A set of events that are both mutually exclusive and exhaustive is known as a partition. The partition that consists of all the fundamental events will be called the fundamental partition. In our example, the two events of selecting a woman and selecting a man form a partition, and the fundamental events associated with each of the 1073 personal selections form the fundamental partition.

A partition consisting of a small number of events, some of which may correspond to many symbols, is known as a coarse-grained partition whereas a partition with many events is a fine-grained partition. The fundamental partition is as fine-grained as any. The partition consisting of the universal event and the null event is as coarse-grained as any.

Although we have described events as though there is always a fundamental partition, in practice this partition need not be used.