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44 CHAPTER 2 PROBABILITY

Total Probability
Rule (two events) For any events A and B,

P1B2 P1B ¨ A2 P1B ¨ A¿2 P1B ƒ A2P1A2 P1B ƒ A¿2P1A¿2 (2-7)

EXAMPLE 2-21 Consider the contamination discussion at the start of this section. Let F denote the event
that the product fails, and let H denote the event that the chip is exposed to high levels of
contamination. The requested probability is P(F), and the information provided can be rep-
resented as

P1F ƒ H2 0.10 and P1F ƒ H¿2 0.005
P1H2 0.20 and P1H¿2 0.80

From Equation 2-7,

P1F2 0.1010.202 0.00510.802 0.0235

which can be interpreted as just the weighted average of the two probabilities of failure.

The reasoning used to develop Equation 2-7 can be applied more generally. In the devel-
opment of Equation 2-7, we only used the two mutually exclusive A and . However, the fact
A¿
that A ´ A¿ S , the entire sample space, was important. In general, a collection of sets
E , E , p , E k such that E 1 ´ E ´p´ E S is said to be exhaustive. A graphical dis-
2
2
k
1
play of partitioning an event B among a collection of mutually exclusive and exhaustive
events is shown in Fig. 2-15 on page 43.

Total Probability
Rule (multiple Assume E , E , p , E k are k mutually exclusive and exhaustive sets. Then
1
2
events)
P1B2 P1B ¨ E 2 P1B ¨ E 2 p P1B ¨ E 2
1
k
2
2P1E 2 P1B ƒ E 2P1E 2 p P1B ƒ E 2P1E 2 (2-8)
P1B ƒ E 1 1 2 2 k k

EXAMPLE 2-22 Continuing with the semiconductor manufacturing example, assume the following probabili-
ties for product failure subject to levels of contamination in manufacturing:

Probability of Failure Level of Contamination
0.10 High
0.01 Medium
0.001 Low