Question

If P(A) > 0, then the event A is independent of itself if and only if P(A) is
A) 1/3
B) 1/2
C) 1
D) None of these

Answer 1

Hint: The literal meaning of independent events is the events which occur freely of each other. In other words, the occurrence of one event does not affect the occurrence of the other. The probability of occurring of the two events are independent of each other. The events A and B are independent if
\[P(A\cap B)=P(A)P(B)\]

Complete step by step answer:
Now here in the given question we have to find that the event A is independent of itself if and only if.
Considering A is independent of itself we have
\[P(A)=P(A\cap A)\]
We also know that
\[\begin{align}
  & P(A\cap B)=P(A)P(B) \\
 & so,P(A\cap A)=P(A)P(A) \\
\end{align}\]
That implies,
 \[\begin{align}
  & P(A)=P(A\cap A)=P(A)P(A) \\
 & P(A)=P(A)P(A) \\
 & P(A)-P(A)P(A)=0 \\
 & P(A)\left[ 1-P(A) \right]=0 \\
\end{align}\]
 From the above equation we get P(A)=0 and P(A)=1
But from our question we know that P(A) > 0 so P(A) cannot be equal to zero
So here we get P(A)=1.
Therefore, the event A is independent of itself if and only if P(A)=1.

So, the correct answer is “Option A”.

Note: In general the probability of occurrence of an independent event is either one or zero, but here as in the question it is mentioned that the probability should be greater than zero we choose our answer to be one. The intersection of two independent events would be zero as they don’t depend on each other.