1.6 Independence of the events

Example: Drawings with Replacement

Two balls are successively drawn from an urn that contains seven red balls and three black balls. The first ball drawn is put back into the urn after noting its color. Then the second ball is drawn.

B = the first ball drawn is black

A = the second ball drawn is black

P(A) = 3/10

10-balls

7- red 3- blackP(A|B)= 3/10

P(A |B) = P(A)

B occurs does not affect the probability of A occurring

Two events, A and B, are independent相互独立性

Two events A and B are independent if

P(A | B) = P(A) ;or, P(B | A) = P(B) .

Definition1.12--- P18

A

B 1 1 1( ) , ( ) , ( )

2 2 4P A P B P AB

AB( ) ( ) ( ).then P AB P A P B

Example

A and B are independent but not disjoint.

A

B

1 1( ) , ( )

2 2P A P B

( ) ( ) ( ).So P AB P A P B

( ) 0,P AB

,41

)()( BPAP

A and B are disjoint but not independent.

Experiment

   A dresser drawer contains one pair of socks with each of the following colors: blue, brown, red, white and black. Each pair is folded together in a matching set. You reach into the sock drawer and choose a pair of socks without looking. You replace this pair and then choose another pair of socks. What is the

probability that you will choose the red pair of socks both times?

                            

                 

P red 1

5

P red and red 1

25 P red P red

  A card is chosen at random from a deck of 52 cards. It is then replaced and a second card is chosen. What is the probability of choosing a jack and then an eight?

     

                 

               

 

Experiment : 

P jack 4

52

P 8 4

52

P jack and 8 P jack P 8 4 4

52 52

16 1

2704 169

A school survey found that 9 out of 10 students like pizza. If three students are chosen at random with replacement, what is the probability that all three students like pizza?

 

   

       

                    

Experiment

P student 1 likes pizza 9

10

9

109

10

P student 2 likes pizza

P student 3 likes pizza

P student 1 and student 2 and student 3 like pizza

P student 1 likes pizza P student 2 likes pizza P student 3 likes pizza 729

1000

, ,A and B A and B A and B

Note: If A and B are independent then

are also independent.

We can extend the definition to arbitrarily many events:

Exercise:P22-Q8、 9、 12