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The units without evidence of gas leaks or electrical failure showed other types of failure. If this is a representative sample of AC failure, nd the probability (a) That failure involves a gas leak (b) That there is evidence of electrical failure given that there was a gas leak (c) That there is evidence of a gas leak given that there is evidence of electrical failure 2-68. If P1A B2 1, must A B Draw a Venn diagram to explain your answer. 2-69. Suppose A and B are mutually exclusive events. Construct a Venn diagram that contains the three events A, B, and C such that P1A C2 1 and P1B C2 0
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MULTIPLICATION AND TOTAL PROBABILITY RULES Multiplication Rule
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The de nition of conditional probability in Equation 2-5 can be rewritten to provide a general expression for the probability of the intersection of two events. This formula is referred to as a multiplication rule for probabilities.
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2-5 MULTIPLICATION AND TOTAL PROBABILITY RULES
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Multiplication Rule P1A B2 P1B A2P1A2 P1A B2P1B2 (2-6)
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The last expression in Equation 2-6 is obtained by interchanging A and B. EXAMPLE 2-20 The probability that an automobile battery subject to high engine compartment temperature suffers low charging current is 0.7. The probability that a battery is subject to high engine compartment temperature is 0.05. Let C denote the event that a battery suffers low charging current, and let T denote the event that a battery is subject to high engine compartment temperature. The probability that a battery is subject to low charging current and high engine compartment temperature is P1C T 2 P1C T 2P1T 2 0.7 0.05 0.035
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Total Probability Rule
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The multiplication rule is useful for determining the probability of an event that depends on other events. For example, suppose that in semiconductor manufacturing the probability is 0.10 that a chip that is subjected to high levels of contamination during manufacturing causes a product failure. The probability is 0.005 that a chip that is not subjected to high contamination levels during manufacturing causes a product failure. In a particular production run, 20% of the chips are subject to high levels of contamination. What is the probability that a product using one of these chips fails Clearly, the requested probability depends on whether or not the chip was exposed to high levels of contamination. We can solve this problem by the following reasoning. For any event B, we can write B as the union of the part of B in A and the part of B in A . That is, B 1A B2 1A B2
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This result is shown in the Venn diagram in Fig. 2-14. Because A and A are mutually exclusive, A B and A B are mutually exclusive. Therefore, from the probability of the union of mutually exclusive events in Equation 2-2 and the Multiplication Rule in Equation 2-6, the following total probability rule is obtained.
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Figure 2-14 Partitioning an event into two mutually exclusive subsets.
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B = (B E1) (B E2) (B E3) (B E4)
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Figure 2-15 Partitioning an event into several mutually exclusive subsets.
CHAPTER 2 PROBABILITY
Total Probability Rule (two events)
For any events A and B, P1B2 P1B A2 P1B A 2 P1B A2P1A2 P1B 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 represented as P1F H2 P1H2 From Equation 2-7, P1F2 0.1010.202 0.00510.802 0.0235 0.10 0.20 and and P1F H 2 P1H 2 0.005 0.80
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 development of Equation 2-7, we only used the two mutually exclusive A and A . However, the fact that A A S , the entire sample space, was important. In general, a collection of sets E1, E2, p , Ek such that E1 E2 p Ek S is said to be exhaustive. A graphical display 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 events)
Assume E1, E2, p , Ek are k mutually exclusive and exhaustive sets. Then P1B2 P1B E1 2 P1B E1 2P1E1 2 P1B E2 2 p P1B E2 2P1E2 2 P1B Ek 2 p P1B Ek 2P1Ek 2
(2-8)
EXAMPLE 2-22
Continuing with the semiconductor manufacturing example, assume the following probabilities for product failure subject to levels of contamination in manufacturing: