In: Statistics and Probability
A batch of 500 disks contains 5 defectiv disks. Two are selected randonly without replacement, from the batch. Let A and B denote the events that the first and the second selected disks are defective, respectively.
a) Are A and B independents events?
b) If the samplins was done with replacement, would A and B be independent?
Let events .
A = the first disk is defective. and B = the second disk is defective.
For Independent events P( A and B ) = P(A)*P(B)
Part a)
There are 5 defective disks and 495 non defective disks, total 500 disks.
We are selecting 2 diske "without replacement"
P(A) = = 5 / 500
P(A) = 0.01
P(B) = P( both disks are defective ) + P( 1st disk is non defective and 2nd disk is defective )
=
= (0.01*0.008016)+(0.99*0.01002)
P(B) = 0.01
Therefore P(A)*P(B) = 0.01*0.01
P(A)*P(B) = 0.0001
Now we have to find P(A and B) , it means both disks are defective .
P( A and B ) =
P( A and B ) = 0.00008
As P( A and B ) P(A)*P(B) , Events A and B are not independent.
Part b) We are selecting 2 disks with replacement method.
P(A) = 5/500 = 0.01
P(B) = P( both disks are defective ) + P( 1st disk is non defective and 2nd disk is defective )
=
= (0.01*0.01) + (0.99*0.01)
P(B) = 0.01
Therefore P(A)*P(B) = 0.01*0.01 = 0.0001
Now we have to find P( A and B), it means both disks are defective .
P( A and B ) =
P( A and B ) = 0.0001
As P(A and B) = P(A)*P(B) , Events A and B are independent.