Show that if events A and B are independent then 1 events A

Show that if events A and B are independent, then:

1) events A^\' and B are independent;

2) events A^\' and B^\' are independent.

Solution

Given that, events A and B are independent.

That means if event A and B are independent then,

P(A B) = P(A) * P(B) _____( is used for intersection)

From this we have to show that,

i) Events A\' and B are independent.

i.e. we have to prove that ,

P(A\' B) = P(A\') * P(B)

There are two facts :

a) We have , P(A) * P(B) = P(B) * ( 1-P(A\') )

P(A) * P(B) = P(B) - P(B) * P(A\')

b) We have , P(A B) = P(B) - P(A\' B)

Substituting a) and b) into the equation.

P(A B) = P(A) * P(B)

P(B) - P(A\' B) = P(A) * P(B)

subtracting P(B) from both sides,

- P(A\' B) = - P(A\') * P(B)

Dividing by (-1),

P(A\' B) = P(A\') * P(B)

Hence the proof.

ii) Events A\' and B\' are independent.

Here we have to prove that,

P(A\' B\') = P( A\' ) * P( B\' )

We showed in part i) that , P(A\' B) = P(A\') * P(B)

We use similarly for P(B) and P(B\').

Fact1 - P(B) + P(B\') = 1

Fact2 - P(A\' B) + P( A\' B\' ) = P(A\')

We have , from fact1,

P(A\') * P(B) = ( 1-P(B\') ) * P(A\')

P(A\') * P(B) = P(A\') - P(A\') * P(B\') ______ a)

And from fact2 ,

P( A\' B ) = P(A\') - P(A\' B\') _________b)

Substituting a) and b) into the equation,

P( A\' B ) = P(A\') * P(B)

P(A\') - P(A\' B\') = P(A\') - P(A\') * P(B\')

Subtracting P(A\') and divide by (-1),

P( A\' B\' ) = P(A\') * P(B\')

Hence the proof.