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Suppose that S is the sample space of a random experiment that contains a finite number of outcomes or elements . In most probability problems, we are more interested in whether or not an outcome belongs to some set E of outcomes rather than in an individual outcome. For instance, if playing the game of craps, one is usually more interested in the total number of dots that appear when the two dice are thrown than in any particular pair of dots obtained from throwing a pair of dice. The inspector who examines five screws taken “at random” from a box of 100 is not really interested in any particular one of the 75,287,250 different sets of five screws he could have drawn; he is in fact looking for the number of defective screws he gets in the five screws he draws. In other words, he is interested in whether this outcome belongs to the set of outcomes with 0 defectives, or the set with one defective, or the set with two defectives, and so on.

Any set of outcomes in which there might be some particular interest is called an event. The following two examples describe two events.

Example 3.2.7 (Sample space generated by tossing two coins) The event of getting exactly one head in throwing the two coins of Example 3.2.2 consists of the set of two elements from the sample space .

Example 3.2.8 (Sample space for playing cards) Suppose that 13 cards are dealt from a deck of ordinary playing cards. Such a deck has 13 cards of each of four suits, which are spades, clubs, hearts, and diamonds. As mentioned in Example 3.2.4, there are 635,013,559,600 possible hands making up the sample space for this experiment (repetitive operation). Now suppose that we are interested in the number of possible hands (elements in ) that contains exactly 12 spades. It turns out that this event (set) contains 507 elements, or put another way, there are 507 hands of 13 cards that contain exactly 12 spades out of the possible 635,013,559,600 hands when dealing 13 cards from a deck of 52 playing cards.

Schematically, if the set of points inside the rectangle in Figure 3.2.1 represent a sample space S, we may represent an event E by the set of points inside a circle and by the region outside the circle. Such a representation is called a Venn diagram.

Figure 3.2.1 Venn diagram representing events E and .

Events can be described in the language of sets, and the words set and event can be used interchangeably. If E contains no elements, it is called the empty, impossible, or null event and is denoted by . The complement of an event E is the event that consists of all elements in S that are not in E. Note, again, that is an event and that .

Now suppose that there are two events E and F in a sample space . The event consisting of all elements contained in E or F, or both, is called the union of E and F; it is written as

(3.2.1)

Figure 3.2.2 Venn diagram representing events , and .

The event consisting of all elements in a sample space S contained in both E and F is called the intersection of E and F; it is written as

(3.2.2)

Referring to the Venn diagram in Figure 3.2.2, note that if S is represented by the points inside the rectangle, E by the points inside the left‐hand circle, and F by the points inside the right‐hand circle, then is represented by the points in the region not shaded in the rectangle and is represented by the points in the region in which the two circles overlap. Also note (see Figure 3.2.1) that , and .

Example 3.2.9 (Union and intersection) Suppose that S is the set of all possible hands of 13 cards, E is the set of all hands containing five spades, and F is the set of all hands containing six honor cards. An honor card is one of either a ten, Jack, Queen, King, or Ace of any suit. Then, is the set of all hands containing five spades or six honor cards, or both. is the set of all hands containing five spades and six honor cards.

If there are no elements that belong to both E and F, then

(3.2.3)

and the sets E and F are said to be disjoint, or mutually exclusive.

If all elements in E are also contained in F, then we say that E is a subevent of F, and we write

(3.2.4)

This means that if E occurs, then F necessarily occurs. We sometimes say that E is contained in F, or that F contains E, if (3.2.4) occurs.

Example 3.2.10 (Sub events) Let S be the sample space obtained when five screws are drawn from a box of 100 screws of which 10 are defective. If E is the event consisting of all possible sets of five screws containing one defective screw and F is the event consisting of all possible sets of the five screws containing at least one defective, then .

If and , then every element of E is an element of F, and vice versa. In this case, we say that E and F are equal or equivalent events; this is written as

(3.2.5)

The set of elements in E that are not contained in F is called the difference between E and F; this is written as

(3.2.6)

If F is contained in E, then is the proper difference between E and F. In this case, we have

(3.2.7)

Example 3.2.11 (Difference of two events) If E is the set of all possible bridge hands with exactly five spades and if F is the set of all possible hands with exactly six honor cards (10, J, Q, K, A), then is the set of all hands with exactly five spades but not containing exactly six honor cards (e.g. see Figure 3.2.3).

If are several events in a sample space S, the event consisting of all elements contained in one or more of the is the union of written as

(3.2.8)

Figure 3.2.3 Venn diagram representing events , and .

Similarly the event consisting of all elements contained in all is the intersection of written as

(3.2.9)

If for every pair of events (), , from we have that , then are disjoint and mutually exclusive events.

An important result concerning several events is the following theorem.

Theorem 3.2.1 If are events in a sample space S, then and are disjoint events whose union is S.

This result follows by noting that the events and are complement of each other.