Applied Statistics and Probability for Engineers

2-5 MULTIPLICATION AND TOTAL PROBABILITY RULES 45

In a particular production run, 20% of the chips are subjected to high levels of contami-

nation, 30% to medium levels of contamination, and 50% to low levels of contamination.

What is the probability that a product using one of these chips fails? Let

Hdenote the event that a chip is exposed to high levels of contamination

Mdenote the event that a chip is exposed to medium levels of contamination

Ldenote the event that a chip is exposed to low levels of contamination

Then,

This problem is also conveniently solved using the tree diagram in Fig. 2-16.

EXERCISES FOR SECTION 2-5

0.10 1 0.20 2 0.01 1 0.30 2 0.001 1 0.50 2 0.0235

P 1 F 2 P 1 FƒH 2 P 1 H 2 P 1 FƒM 2 P 1 M 2 P 1 FƒL 2 P 1 L 2

0.10(0.20)

= 0.02

0.90(0.20)

= 0.18

0.01(0.30)

= 0.003

0.99(0.30)

= 0.297

0.001(0.50)

= 0.0005

0.999(0.50)

= 0.4995

P(Fail) = 0.02 + 0.003 + 0.0005 = 0.0235

Contamination

0.20 0.50

0.30

High Medium Low

P(FailHigh)

= 0.10

P(Not FailHigh)

= 0.90

P(FailMedium)

= 0.01

P(Not FailMedium)

= 0.99

P(FailLow)

= 0.001

P(Not FailLow)

= 0.999

Figure 2-16 Tree

diagram for

Example 2-22.

2-70. Suppose that and

Determine the following:

(a)

(b)

2-71. Suppose that and

What is P(A)?

2-72. The probability is 1% that an electrical connector that

is kept dry fails during the warranty period of a portable com-

puter. If the connector is ever wet, the probability of a failure

during the warranty period is 5%. If 90% of the connectors are

kept dry and 10% are wet, what proportion of connectors fail

during the warranty period?

2-73. Suppose 2% of cotton fabric rolls and 3% of nylon

fabric rolls contain flaws. Of the rolls used by a manufacturer,

70% are cotton and 30% are nylon. What is the probability

that a randomly selected roll used by the manufacturer con-

tains flaws?

P 1 B 2 0.8.

P 1 AƒB 2 0.2, P 1 AƒB¿ 2 0.3,

P 1 A¿ ̈B 2

P 1 A ̈B 2

P 1 AƒB 2 0.4 P 1 B 2 0.5. 2-74. In the manufacturing of a chemical adhesive, 3% of

all batches have raw materials from two different lots. This

occurs when holding tanks are replenished and the remaining

portion of a lot is insufficient to fill the tanks.

Only 5% of batches with material from a single lot require

reprocessing. However, the viscosity of batches consisting of

two or more lots of material is more difficult to control, and

40% of such batches require additional processing to achieve

the required viscosity.

Let Adenote the event that a batch is formed from

two different lots, and let Bdenote the event that a lot

requires additional processing. Determine the following

probabilities:

(a) (b)

(c) (d)

(e) (f)

(g)P 1 B 2

P 1 A ̈B 2 P 1 A ̈B¿ 2

P 1 BƒA 2 P 1 BƒA¿ 2

P 1 A 2 P 1 A¿ 2

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