Applied Statistics and Probability for Engineers

2-4 CONDITIONAL PROBABILITY 39

Therefore, can be interpreted as the relative frequency of event Bamong the trials that

produce an outcome in event A.

EXAMPLE 2-17 Again consider the 400 parts in Table 2-3. From this table

Note that in this example all four of the following probabilities are different:

Here, P(D) and are probabilities of the same event, but they are computed under two

different states of knowledge. Similarly, P(F) and are computed under two different

states of knowledge.

The tree diagram in Fig. 2-13 can also be used to display conditional probabilities. The

first branch is on surface flaw. Of the 40 parts with surface flaws, 10 are functionally defec-

tive and 30 are not. Therefore,

Of the 360 parts without surface flaws, 18 are functionally defective and 342 are not. Therefore,

Random Samples from a Batch

Recall that to select one item randomly from a batch implies that each item is equally likely.

If more than one item is selected, randomlyimplies that each element of the sample space is

equally likely. For example, when sample spaces were presented earlier in this chapter, sam-

pling with and without replacement were defined and illustrated for the simple case of a batch

with three items {a,b,c}. If two items are selected randomly from this batch without replace-

ment, each of the six outcomes in the ordered sample space

has probability. If the unordered sample space is used, each of the three outcomes in

{{a,b}, {a, c}, {b, c}} has probability 1

3.

1
6

Swithout 5 ab, ac, ba, bc, ca, cb 6

P 1 DƒF¿ 2  342

360 and P 1 D¿ƒF¿ 2  18
360

P 1 DƒF 2  10

40 and P 1 D¿ƒF 2  30
40

P 1 FƒD 2

P 1 DƒF 2

P 1 D 2  28

400 P 1 DƒF 2  10
40

P 1 F 2  40

400 P 1 FƒD 2  10
28

P 1 DƒF 2 P 1 D ̈F 2
P 1 F 2 

10

400 ^

40

400



10

40

P 1 BƒA 2

Surface flaw

No Yes

No Yes No Yes

Defective

360

400

40

400

342

360

18

360

30

40

10

Figure 2-13 Tree 40

diagram for parts

classified

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