Page 66 - Applied Statistics And Probability For Engineers

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2-6 INDEPENDENCE 49

not dependent on the characteristics of any of the other wafers. If 15 wafers are analyzed, what
is the probability that no large particles are found?
Let E denote the event that the ith wafer contains no large particles, i 1, 2, p , 15.
i
Then, P1E 2 0.99. The probability requested can be represented as P1E ¨ E ¨ p ¨ E 2.
15
i
2
1
From the independence assumption and Equation 2-10,
15
p
¨ E ¨ ¨ E 2 P1E 2 P1E 2 p P1E 2 0.99 0.86
P1E 1 2 15 1 2 15
EXAMPLE 2-27 The following circuit operates only if there is a path of functional devices from left to right.
The probability that each device functions is shown on the graph. Assume that devices fail
independently. What is the probability that the circuit operates?

0.95

a b
0.95

Let T and B denote the events that the top and bottom devices operate, respectively. There
is a path if at least one device operates. The probability that the circuit operates is

P1T or B2 1 P31T or B2¿4 1 P1T¿ and B¿2

a simple formula for the solution can be derived from the complements T¿ and B¿. From the
independence assumption,

2
P1T¿ and B¿2 P1T¿2P1B¿2 11 0.952 0.05 2
so

2
P1T or B2 1 0.05 0.9975
EXAMPLE 2-28 The following circuit operates only if there is a path of functional devices from left to right.
The probability that each device functions is shown on the graph. Assume that devices fail
independently. What is the probability that the circuit operates?

0.9
0.95
a 0.9 0.99 b

0.95
0.9