Applied Statistics and Probability for Engineers

2-4 CONDITIONAL PROBABILITY 37

Tool 1

roundness conforms

yes no

surface finish yes 200 1

conforms no 4 2

Tool 2

roundness conforms

yes no

surface finish yes 145 4

conforms no 8 6

(a) If a shaft is selected at random, what is the probability that

the shaft conforms to surface finish requirements or to

roundness requirements or is from Tool 1?

(b) If a shaft is selected at random, what is the probability that

the shaft conforms to surface finish requirements or does

not conform to roundness requirements or is from Tool 2?

(c) If a shaft is selected at random, what is the probability that

the shaft conforms to both surface finish and roundness

requirements or the shaft is from Tool 2?

(d) If a shaft is selected at random, what is the probability that

the shaft conforms to surface finish requirements or the

shaft is from Tool 2?

2-4 CONDITIONAL PROBABILITY

A digital communication channel has an error rate of one bit per every thousand transmitted.

Errors are rare, but when they occur, they tend to occur in bursts that affect many consecutive

bits. If a single bit is transmitted, we might model the probability of an error as 11000.

However, if the previous bit was in error, because of the bursts, we might believe that the

probability that the next bit is in error is greater than 11000.

In a thin film manufacturing process, the proportion of parts that are not acceptable is 2%.

However, the process is sensitive to contamination problems that can increase the rate of parts

that are not acceptable. If we knew that during a particular shift there were problems with the

filters used to control contamination, we would assess the probability of a part being unac-

ceptable as higher than 2%.

In a manufacturing process, 10% of the parts contain visible surface flaws and 25% of the

parts with surface flaws are (functionally) defective parts. However, only 5% of parts without

surface flaws are defective parts. The probability of a defective part depends on our knowl-

edge of the presence or absence of a surface flaw.

These examples illustrate that probabilities need to be reevaluated as additional informa-

tion becomes available. The notation and details are further illustrated for this example.

Let Ddenote the event that a part is defective and let Fdenote the event that a part has a

surface flaw. Then, we denote the probability of Dgiven, or assuming, that a part has a sur-

faceflaw as. This notation is read as the conditional probability of Dgiven F,and it

is interpreted as the probability that a part is defective, given that the part has a surface flaw.

Because 25% of the parts with surface flaws are defective, our conclusion can be stated as

. Furthermore, because denotes the event that a part does not have a surface
flaw and because 5% of the parts without surface flaws are defective, we have that
P 1 DƒF¿ 2 0.05. These results are shown graphically in Fig. 2-12.

P 1 DƒF 2 0.25 F¿

P 1 DƒF 2

5% defective

P(DF’) = 0.05

F’ = parts without

surface flaws

25%

defective

P( DF) = 0.25

F = parts with

surface flaws

Figure 2-12

Conditional probabili-

ties for parts with

surface flaws.

c 02 .qxd 5/10/02 1:07 PM Page 37 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files: