Likelihood  (Page 4/4)

One of the advantages of this formulation of an event as a subset of the basic set of possible outcomes is that we can use elementary set theory as an aid toformulation. And tools, such as Venn diagrams and indicator functions for studying event combinations, provide powerful aids to establishing and visualizing relationshipsbetween events. We formalize these ideas as follows:

• Let $\Omega$ be the set of all possible outcomes of the basic trial or experiment. We call this the basic space or the sure event , since if the trial is carried out successfully the outcome will be in $\Omega$ ; hence, the event $\Omega$ is sure to occur on any trial. We must specify unambiguously what outcomes are “possible.” In flippinga coin, the only accepted outcomes are “heads” and “tails.” Should the coin stand on its edge, say by leaning against a wall, we would ordinarily consider that to be the result ofan improper trial.
• As we note above, each outcome may have several characteristics which are the basis for describing events. Suppose we are drawing a single card from an ordinary deckof playing cards. Each card is characterized by a “face value” (two through ten, jack, queen, king, ace) and a “suit” (clubs, hearts, diamonds, spades). An ace is drawn (the event ACEoccurs) iff the outcome (card) belongs to the set (event) of four cards with ace as face value. A heart is drawn iff the card belongs to the set of thirteen cards with heart as suit. Nowit may be desirable to specify events which involve various logical combinations of the characteristics. Thus, we may be interested in the event the face value is jack or king and the suit is heart or spade. The set for jack or king is represented by the union $J\cup K$ and the set for heart or spade is the union $H\cup S$ . The occurrence of both conditions means the outcome is in the intersection (common part) designated by $\cap$ . Thus the event referred to is
  $$E=(J\cup K)\cap (H\cup S)$$
  The notation of set theory thus makes possible a precise formulation of the event $E$ .
• Sometimes we are interested in the situation in which the outcome does not have one of the characteristics. Thus the set of cards which does not have suit heart is theset of all those outcomes not in event $H$ . In set theory, this is the complementary set (event) $H^c$ .
• Events are mutually exclusive iff not more than one can occur on any trial. This is the condition that the sets representing the events are disjoint (i.e., haveno members in common).
• The notion of the impossible event is useful. The impossible event is, in set terminology, the empty set $\varnothing$ . Event $\varnothing$ cannot occur, since it has no members (contains no outcomes). One use of $\varnothing$ is to provide a simple way of indicating that two sets are mutually exclusive. To say $AB=\varnothing$ (here we use the alternate $AB$ for $A\cap B$ ) is to assert that events $A$ and $B$ have no outcome in common, hence cannot both occur on any given trial.
• Set inclusion provides a convenient way to designate the fact that event $A$ implies event $B$ , in the sense that the occurrence of $A$ requires the occurrence of $B$ . The set relation $A\subset B$ signifies that every element (outcome) in $A$ is also in $B$ . If a trial results in an outcome in $A$ (event $A$ occurs), then that outcome is also in $B$ (so that event $B$ has occurred).

The language and notaton of sets provide a precise language and notation for events and their combinations. We collect below some useful facts about logical (often called Boolean) combinations ofevents (as sets). The notion of Boolean combinations may be applied to arbitrary classes of sets. For this reason, it is sometimes useful to use an index set to designate membership. We say the index J is countable if it is finite or countably infinite; otherwise it is uncountable . In the following it may be arbitrary.

For example, if $J=\{1,2,3\}$ then $\{A_i:i\in J\}$ is the class $\{A_1,A_2,A_3\}$ , and

If $J=\{1,2,\cdots \}$ then $\{A_i:i\in J\}$ is the sequence $\{A_1:1\le i\}$ . and

If event E is the union of a class of events, then event E occurs iff at least one event in the class occurs. If F is the intersection of a class of events, then event F occurs iff all events in the class occur on the trial.

The role of disjoint unions is so important in probability that it is useful to have a symbol indicating the union of a disjoint class. We use the big Vto indicate that the sets combined in the union are disjoint. Thus, for example, we write

Two important patterns in set theory known as DeMorgan's rules are useful in the handling of events. For an arbitrary class $\{A_i:i\in J\}$ of events,

An outcome is not in the union (i.e., not in at least one) of the A _i iff it fails to be in all A _i , and it is not in the intersection (i.e. not in all) iff it fails to be in at least one of the A _i .