Page 57 - Applied statistics and probability for engineers

Section 2-3/Addition Rules 35

5"#-& t 2E-1 Average Frequencies and Operations in TPC-C
Transaction Frequency Selects Updates Inserts Deletes Nonunique Joins
Selects
New order 43 23 11 12 0 0 0
Payment 44 4.2 3 1 0 0.6 0
Order status 4 11.4 0 0 0 0.6 0
Delivery 5 130 120 0 10 0 0
Stock level 4 0 0 0 0 0 1

(
(
(a) P A∩ B) (b) P A( ′) (c) P A∪ B) 2-79. Similar to the hospital schedule in Example 2-11, suppose
(
(
(d) P A∪ ) (e) P A′ ∩ B′) that an oper ating room needs to schedule three knee, four hip,
B′
and ive shoulder surgeries. Assume that all schedules are equally

2-75. A Web ad can be designed from four different colors, likely. Determine the probability for each of the following:
three font types, i ve font sizes, three images, and i ve text (a) All hip surgeries are completed before another type of surgery.
phrases. A speciic design is randomly generated by the Web (b) The schedule begins with a hip surgery.

server when you visit the site. If you visit the site i ve times, (c) The irst and last surgeries are hip surgeries.

what is the probability that you will not see the same design? (d) The irst two surgeries are hip surgeries.

2-76. Consider the hospital emergency room data in Example
2-8. Let A denote the event that a visit is to hospital 4, and let B 2-80. Suppose that a patient is selected randomly from the
denote the event that a visit results in LWBS (at any hospital). those described in Exercise 2-57. Let A denote the event that
Determine the following probabilities. the patient is in the group treated with interferon alfa, and let
(
(
(a) P A∩ B) (b) P A( )′ (c) P A∪ B) B denote the event that the patient has a complete response.
(
(
(d) P A∪ ) (e) P A′ ∩ B′) Determine the following probabilities.
B′
2-77. Consider the well failure data in Exercise 2-53. Let A (a) P A( ) (b) P B( )
denote the event that the geological formation has more than (c) P A( ∩ B) (d) P A( ∪ B) (e) P A( ′ B)
1000 wells, and let B denote the event that a well failed. Deter- 2-81. A computer system uses passwords that contain
mine the following probabilities. exactly eight characters, and each character is one of 26 low-
(
(
(a) P A∩ B) (b) P A( )′ (c) P A∪ B) ercase letters (a–z) or 26 uppercase letters (A–Z) or 10 inte-
(d) P A∪ ) (e) P A′ ∩ B′) gers (0–9). Let Ω denote the set of all possible passwords,
(
(
B′
and let A and B denote the events that consist of passwords
2-78. Consider the bar code in Example 2-12. Suppose that all
40 codes are equally likely (none is held back as a delimiter). with only letters or only integers, respectively. Suppose that
Determine the probability for each of the following: all passwords in Ω are equally likely. Determine the probabil-
(a) A wide space occurs before a narrow space. ity of each of the following:
(b) Two wide bars occur consecutively. (a) A (b) B
(c) Two consecutive wide bars are at the start or end. (c) A password contains at least 1 integer.
(d) The middle bar is wide. (d) A password contains exactly 2 integers.
2-3 Addition Rules
Joint events are generated by applying basic set operations to individual events. Unions of events,
such as A∪ B; intersections of events, such as A∩ B; and complements of events, such as A′—are
commonly of interest. The probability of a joint event can often be determined from the probabili-
ties of the individual events that it comprises. Basic set operations are also sometimes helpful in
determining the probability of a joint event. In this section, the focus is on unions of events.
Example 2-19 Semiconductor Wafers Table 2-1 lists the history of 940 wafers in a semiconductor manu-
facturing process. Suppose that 1 wafer is selected at random. Let H denote the event that the
wafer contains high levels of contamination. Then, P H( ) = 358 940.
/
(
/
Let C denote the event that the wafer is in the center of a sputtering tool. Then, P C( ) = 626 940 . Also, P H ∩ C) is the
probability that the wafer is from the center of the sputtering tool and contains high levels of contamination. Therefore,