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

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.

3.3 Concepts of Probability

      Suppose that a sample space S, consists of a finite number, say m, of elements , so that the elements are such that for all and also represent an exhaustive list of outcomes in S, so that . If the operation whose sample space is S is repeated a large number of times, some of these repetitions will result in , some in , and so on. (The separate repetitions are often called trials.) Let be the fractions of the total number of trials resulting in , respectively. Then, Скачать книгу