Page 58 - Applied statistics and probability for engineers

36 Chapter 2/Probability

(
P H ∩ C) = 112 940
/
The event H ∪ C is the event that a wafer is from the center of the sputtering tool or contains high levels of contamination
(
/
(or both). From the table, P H( ∪ C) = 872 940 . An alternative calculation of P H ∪ C) can be obtained as follows. The
112 wafers in the event H ∩ C are included once in the calculation of P H( ) and again in the calculation of P C( ). Therefore,
(
P H ∪ C) can be determined to be
(
P H) + (
P H ⎟
P H ∪ C) = ( P C) − ( C)
+
/
/
/
= 358 940 626 940 −112 940 = 872 940
/
Practical Interpretation: To better understand the sources of contamination, yield from defferent locations on wafers
are routinely aggregated.
5 2-1 Wafers in Semiconductor Manufacturing Classifi ed by
Contamination and Location
Location in Sputtering Tool
Contamination Center Edge Total
Low 514 68 582
High 112 246 358
Total 626 314

(
The preceding example illustrates that the probability of A or B is interpreted as P A∪ B)
and that the following general addition rule applies.
Probability of
P A) + (
a Union ( B) = ( P B) − ( B)
P A∪ P A⎟ (2-5)

Example 2-20 Semiconductor Wafers and Location The wafers in Example 2-19 were further classii ed by
the degree of contamination. Table 2-2 shows the proportion of wafers in each category. What is the
probability that a wafer was either at the edge or that it contains four or more particles? Let E 1 denote the event that a wafer
contains four or more particles, and let E 2 denote the event that a wafer was at the edge.
(
.
The requested probability is P E 1 ∪ ). Now, P E 1 ( ) = 0 15 and P E 2 ( ) = 0 28. Also, from the table,
.
E 2
(
.
P E 1 ∩ ) = 0 04. Therefore, using Equation 2-1, we i nd that
E 2
(
0 15 0 28 0 04 = .
P E 1 ∪ ) = . + . − . 0 39
E 2
5"#-& t 2-2 Wafers Classifi ed by Contamination and Location
Number of
Contamination
Particles Center Edge Totals
0 0.30 0.10 0.40
1 0.15 0.05 0.20
2 0.10 0.05 0.15
3 0.06 0.04 0.10
4 0.04 0.01 0.05
5 or more 0.07 0.03 0.10
Totals 0.72 0.28 1.00