Applied Statistics and Probability for Engineers

The conditional probabilityof an event Bgiven an event A, denoted as , is

(2-5)

for .P 1 A 2  0

P 1 BƒA 2 P 1 A ̈B 2
P 1 A 2

P 1 BƒA 2

Definition

38 CHAPTER 2 PROBABILITY

EXAMPLE 2-16 Table 2-3 provides an example of 400 parts classified by surface flaws and as (functionally)

defective. For this table the conditional probabilities match those discussed previously in this

section. For example, of the parts with surface flaws (40 parts) the number defective is 10.

Therefore,

and of the parts without surface flaws (360 parts) the number defective is 18. Therefore,

In Example 2-16 conditional probabilities were calculated directly. These probabilities

can also be determined from the formal definition of conditional probability.

P 1 DƒF¿ 2  18
360 0.05

P 1 DƒF 2  10
40 0.25

Table 2-3 Parts Classified

Surface Flaws

Yes (event F) No Total

Defective Yes (event D)10 18 38

No 30 342 362

Total 40 360 400

This definition can be understood in a special case in which all outcomes of a random exper-

iment are equally likely. If there are ntotal outcomes,

Also,

Consequently,

P 1 A ̈B 2
P 1 A 2 

number of outcomes in A ̈B

number of outcomes in A

P 1 A ̈B 2  1 number of outcomes in A ̈B 2
n

P 1 A 2  1 number of outcomes in A 2
n

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