Question

if A and B are two independent events then prove that A and B' are independent events.​

Answer 1

Given :-

• A and B sre independent events.

To Prove :-

• A and B' are independent.

Concept Used :-

In order to prove that A and B' are independent, it is enough to prove that

$\red{\boxed{\sf \: P(A \cap \: B') \: = \: P(A) \: \: P(B')}}$

Solution :-

Given that

• A and B are independent events.

$\rm :\implies\:P(A \: \cap \: B) = \: P(A) \: \: P(B) - - - (1)$

Consider,

$\rm :\longmapsto\:P(A \: \cap \: B')$

$\sf \: = \: \: P(A) \: - \: P(A \: \cap \: B)$

$\sf \: = \: \: P(A) \: - \: P(A )P( B)$

$\sf \: = \: \: P(A) \: (1- \: P( B))$

$\sf \: = \: \: P(A) \: P( B')$

${\boxed{\boxed{\bf{Hence, Proved}}}}$

Additional Information :-

$\red{\boxed{\sf \:P(A) + P(A') = 1}}$

$\red{\boxed{\sf \:P(A \: \cap \: B') = P(A) - P(A \: \cap \: B)}}$

$\red{\boxed{\sf \:P(A' \: \cap \: B) = P(B) - P(A \: \cap \: B)}}$

$\red{\boxed{\sf \:P(A' \: \cap \: B') = 1 - P(A \: \cup \: B)}}$

$\red{\boxed{\sf \:P(A' \: \cup \: B') = 1 - P(A \: \cap \: B)}}$

$\red{\boxed{\sf \:P(A |B) = \dfrac{P(A \: \cap \: B)}{P(B)}\: = \dfrac{n(A \: \cap \: B)}{n(B)}}}$