Applied Statistics and Probability for Engineers

2-5 MULTIPLICATION AND TOTAL PROBABILITY RULES 43

The last expression in Equation 2-6 is obtained by interchanging Aand 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 Cdenote the event that a battery suffers low charging current, and let Tdenote 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

2-5.2 Total Probability Rule

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 contamina-

tion 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 Bas the union of the part of Bin Aand the part of Bin. That is,

This result is shown in the Venn diagram in Fig. 2-14. Because Aand are mutually exclu-

sive, and 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 ruleis obtained.

A ̈B A¿ ̈B

A¿

B 1 A ̈B 2 ́ 1 A¿ ̈B 2

A¿

P 1 C ̈T 2 P 1 CƒT 2 P 1 T 2 0.70.050.035

AA'

B

B ∩ A

B ∩ A'

Figure 2-14 Partitioning

an event into two mutually

exclusive subsets.

E 1

B ∩ E 1

E (^2) E
3
E 4
B ∩ E 2
B ∩ E 3
B ∩ E 4
B = (B ∩ E 1 ) ∪ (B ∩ E 2 ) ∪ (B ∩ E 3 ) ∪ (B ∩ E 4 )
Figure 2-15 Partitioning an event into
several mutually exclusive subsets.
P 1 A ̈B 2 P 1 BƒA 2 P 1 A 2 P 1 AƒB 2 P 1 B 2 (2-6)
Multiplication Rule
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