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2-5 MULTIPLICATION AND TOTAL PROBABILITY RULES 43

Multiplication Rule
P1A ¨ B2 P1B ƒ A2P1A2 P1A ƒ B2P1B2 (2-6)

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 ƒ T 2P1T 2 0.7 0.05 0.035

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
A
B, we can write B as the union of the part of B in A and the part of B in . That is,
¿
B 1A ¨ B2 ´ 1A¿¨ B2

This result is shown in the Venn diagram in Fig. 2-14. Because A and A¿ are mutually exclu-
sive, 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.

A A' E 1 E
2
E 3
B ∩ E E
B ∩ A 1 B ∩ E 4
B ∩ A' 2
B ∩ E 3
B ∩ E
B 4
Figure 2-14 Partitioning B = (B ∩ E ) ∪ (B ∩ E ) ∪ (B ∩ E ) ∪ (B ∩ E )
4
3
1
2
an event into two mutually
Figure 2-15 Partitioning an event into
exclusive subsets.
several mutually exclusive subsets.