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(FailHigh)
= 0.10
P(Not FailHigh)
= 0.90
P(FailMedium)
= 0.01
P(Not FailMedium)
= 0.99
P(FailLow)
= 0.001
P(Not FailLow)
= 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
c 02 .qxd 5/10/02 1:07 PM Page 45 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files: