Page 59 - Applied statistics and probability for engineers

Section 2-3/Addition Rules 37

What is the probability that a wafer contains less than two particles or that it is both at the edge and contains more than
four particles? Let E 1 denote the event that a wafer contains less than two particles, and let E 2 denote the event that a wafer
(
.
is both at the edge and contains more than four particles. The requested probability is P E 1 ∪ ). Now, P E 1 ( ) = 0 60
E 2
and P E 2 ( ) = 0 03. Also, E 1 and E 2 are mutually exclusive. Consequently, there are no wafers in the intersection and
.
(
P E 1 ∩ ) = 0. Therefore,
E 2
(
+
.
.
.
P E 1 ∪ ) = 0 60 0 03 = 0 63
E 2
Recall that two events A and B are said to be mutually exclusive if A∩ B = ∅. Then,
(
P A∩ B) = 0, and the general result for the probability of A∪ B simpliies to the third axiom

of probability.
If A and B are mutually exclusive events,
P A∪ B) = P A) + P B) (2-6)
(
(
(
Three or More Events
(
More complicated probabilities, such as P A∪ ∪ C), can be determined by repeated use of
B
Equation 2-5 and by using some basic set operations. For example,
(
P C) − ( ⎡
( ⎡
P A∪
P A∪ ∪ C) = P A∪ B)∪ ⎤ = ( B) + ( P AÐ B)∩ ⎤ ⎦
C
B
C
⎣
⎦
⎣
(
Upon expanding P A∪ B) by Equation 2-5 and using the distributed rule for set operations to
( ⎡
simplify P A∪ B)∩ ⎤, we obtain
C
⎣
⎦
(
P A) + (
P C) − ( ⎡
P A∪ ∪ C) = ( P B) − ( B) + ( P ⎣ A⎟ C)∪( B ∩ C)⎤ ⎦
P A⎟
B
P C) − (
= P A ( ) + ( P( B) + ( ⎡ ⎣ P A⎟ C) + ( C) − ( B∩ C)⎤ ⎦
P B) − P A∩
P B ∩
P A⎟
P B) + (
P A⎟ B) − (
= P A ( ) + ( P C) − ( B P A⎟ C) − ( C) + ( B C)
P B ⎟
P A∩ ∩
We have developed a formula for the probability of the union of three events. Formulas can
be developed for the probability of the union of any number of events, although the formulas
become very complex. As a summary, for the case of three events,
P A) + (
(
P C) − (
P A∪ ∪ C) = ( P B) + ( P A⎟ B)
B
P
− (A ⎟ ) − (B ⎟ ) + (A ∩ ∩ ) (2-7)
P
C
C
B
C
P
Results for three or more events simplify considerably if the events are mutually exclusive.
In general, a collection of events, E ,E , … ,E , is said to be mutually exclusive if there is no
2
1
k
overlap among any of them. The Venn diagram for several mutually exclusive events is shown
in Fig. 2-12. By generalizing the reasoning for the union of two events, the following result
can be obtained:
E 1
E 2 E 4
FIGURE 2-12 Venn diagram of four E 3
mutually exclusive events.