Show probability is NOT 0.5 for independent events?

Question

Let $Ω$ denote the sample space.

Let $A ⊆ Ω$ such that the events $A$ and $B$ are statistically independent for all $B ⊆ Ω$. Show that $P(A) ≠ 0.5$.

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I have no idea how to go about this. I'm not even sure I understand the information given in the question! ($A$ and $B$ are independent for all elements of $B$ that are in the sample space?) It sounds to me like conditional probability might be involved, so $P(A|B) = P(A)$, but even if that's right, I don't know where to take it.

Any hints? Thanks.

Answer 1

Remember that an event is a subset of the sample space, not an element. The probability measure associates a number to every subset of $\Omega$, subject to additivity and $P(\Omega)=1$.

If every subset of $\Omega$ is independent to $A$, then $P(A\cap B)=P(A)P(B)$. But that "for all" is a bit troubling, because it doesn't say "for all $B\ne A$", so this means that $A$ is independent to itself! (Is that even possible?) Following this line of thought:

$$P(A)=P(A\cap A)=P(A)P(A)=P(A)^2$$

The only solutions to $x=x^2$ are $x=0$ or $x=1$, so either $A$ always happens or it never does; $P(A)=0.5$ is not an option.

Answer 2

Well, if it is true for all $B \subseteq \Omega$ this means that for $B = A$ one finds $$ P[A] = P[A \cap A] = P[A]P[A] = P[A]^2, $$ which means that $P[A] \in \{0,1\}$. So $P[A] \neq 0.5$.