Let's consider this example.

Example 12 : In a supermarket, there are 100 shirts, 30 produced by company X, 70 produced by company Y. 80% shirts of company X are classified as class A, whereas only 30% shirts of company Y are classified as class A. This situation is illustrated in Fig. 1.

Fig. 1 Classification of shirts in the supermarket

Therefore, in this supermarket, there are 28 + 24 = 52 shirts classified as class A.

Now, draw randomly a shirt. What is the probability that this shirt belongs to class A?

There are three cases :

• If we don't know any thing about the company that produced this shirt, the probability is 0,52.
• If we know that this shirt is produced by company X, the probability is 0,80.
• If we know that this shirt is produced by company Y, the probability is 0,30.

So the probability of an event (shirt belongs to class A) changes depend on the occurrence of another event (company that produced shirt).

 

In the example above, we see that the probability of event E changes when we have the information about another event F, or when another event F happens. This type of probability known as conditional probability, symbolized as `P("E|F")`, and determined by:

`P("E|F")=(P("E"nn"F"))/(P("F"))`

(6)

Formula (6) is also known as Bayes theorem, which is applied a lot.

We can transpose (6) to another form :

`P("E"nn"F")=P("F")\ P("E|F")`(7)

(7) is also referred as multiplication rule of probability.

If we examine Example 12 then:

• E : event "shirt belonging to class A" : `P("E")=0,52`
• F : event "shirt produced by company X" : `P("F")=0,30`
• E∩F : event "shirt belonging to class A and produced by company X" : `P("E"nn"F")=0,24`
• E|F : event "shirt belongs to class A in the condition that it produced by company X": `P("E|F")=0,80`

So :   `(P("E"nn"F"))/(P("F"))=(0,24)/(0,3)=0,8=P("E|F")`

 

Two events are independent when the probability of an event does not depend on the occurrence of other event. In this case:

    `P("E|F")=P("E")`   and   `P("F|E")=P("F")`   and   `P("E"nn"F")=P("E")\ P("F")`

In Example 12, if 80% shirts of company Y classified as class A, then the new situation is illustrated in Fig. 2.

Fig. 2 The new classification of shirt in the supermarket

Now :   `P("E")=(24 + 56)/100=0,80=P("E|F")`

In this case, two events "shirt belonging to class A" and "shirt produced by company X" are independent.