Applied Statistics and Probability for Engineers

2-6 INDEPENDENCE 47

EXAMPLE 2-24 The information in Table 2-3 related surface flaws to functionally defective parts. In that case,

we determined that and Suppose that the

situation is different and follows Table 2-4. Then,

That is, the probability that the part is defective does not depend on whether it has surface

flaws. Also,

so the probability of a surface flaw does not depend on whether the part is defective.

Furthermore, the definition of conditional probability implies that

but in the special case of this problem

The preceding example illustrates the following conclusions. In the special case that

we obtain

and

These conclusions lead to an important definition.

P 1 AƒB 2 

P 1 A ̈B 2

P 1 B 2



P 1 A 2 P 1 B 2

P 1 B 2

P 1 A 2

P 1 A ̈B 2 P 1 BƒA 2 P 1 A 2 P 1 B 2 P 1 A 2

P 1 B 0 A 2 P 1 B 2 ,

P 1 F ̈D 2 P 1 D 2 P 1 F 2 

2

40



2

20



1

200

P 1 F ̈D 2 P 1 DƒF 2 P 1 F 2

P 1 FƒD 2  2

20 0.10 and P 1 F 2  40
400 0.10

P 1 DƒF 2  2

40 0.05 and P 1 D 2  20
400 0.05

P 1 DƒF 2  10
40 0.25 P 1 D 2  28
400 0.07.

Table 2-4 Parts Classified

Surface Flaws

Yes (event F) No Total

Defective Yes (event D) 2 18 20

No 38 342 380

Total 40 360 400

Two events are independentif any one of the following equivalent statements is true:

(1)

(2)

(3) P 1 A ̈B 2 P 1 A 2 P 1 B (^2) (2-9)
P 1 BƒA 2 P 1 B 2
P 1 AƒB 2 P 1 A 2
Definition
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