Independent probabilities

Question

Given 2 independent probabilities A and B, I have to prove that A' and B are independent and also that A' and B' are independent. I have thought of using the formulas P(A ∩ B) = P(A)*P(B) and P(A|B) = $\frac{P(A ∩ B)}{P(B)}$, but I still haven't found a solution.

Answer 1

You can check using a venn diagram that the region represented by $A' \cap B$ can be written as $B - (A \cap B)$

So,

$$P(A' \cap B) = P(B) - P(A \cap B) $$

Also using the fact that $A$ and $B$ are independent ($P(A \cap B) = P(A) \times P(B) $)

$$P(A' \cap B) = P(B) - P(A)P(B) $$

$$=(P(B) )(1-P(A))$$

$$P(A' \cap B) = P(B) \times P(A') $$

Thus its proved that they are independent.

You can do the same (or similar) in the next problem by using the De morgan's Law: $A' \cap B' = (A \cup B)'$