Bhisham C. Gupta - Statistics and Probability with Applications for Engineers and Scientists Using MINITAB, R and JMP

Schematically, if the set of points inside the rectangle in Figure 3.2.1represent 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 3.2.1Venn 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 3.2.2Venn 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