Question

Let A, B, C be three events such that A and B are independent and $P\left(C\right)=0$, then events A, B, C are

• A. independent
• B. pairwise independent but not totally independent
• C. $P\left(A\right)=P\left(B\right)=P\left(C\right)$
• D. none of these

Answer 1

Answer: (A)

independent

Given three events $A,B,C.$

A and B are independent $\xi$ $P\left(C\right)=0.$

A term 'independent event' means probability of events. They have a common outcomes and each event is non-empty.

As A and B are independent

i.e, $P\left(A\right)\ne 0$ and $P\left(B\right)\ne 0.$

For $P\left(C\right)$ to be independent,

$\Rightarrow P(A\cap B)=P\left(A\right).P\left(B\right)$ $P(A\cap B\cap C)=P\left(A\right).P\left(B\right).P\left(C\right)$

$\Rightarrow P(A\cap C)=P\left(A\right).P\left(C\right)$

$\Rightarrow P(B\cap C)=P\left(B\right).P\left(C\right)$

If all the given events are true, then A,B,C are independent.

Hence, the answer is independent.