$P(A)=a$ and $X=1_{\{A\}}$ what is Var(X)

Question

$P(A)=a$ and $X=1_{\{A\}}$ what is $Var(X)$

So what I did:

$$Var(X)=E[X^2]-E[X]^2$$

now $E[X]=P(A)=a$ so we have

$$Var(X)=E[X^2]-a^2$$

Does $E[X^2]$ also equal $a^2$?

making the answer $Var(X)=0$?

Answer 1

Expectation of $X^2$ is $1^2\cdot a+0^2(1-a)=a$, giving the variance $a-a^2=a(1-a)$. Note that $X$ is a Bernoulli trial so its variance was expected to be $pq=p(1-p)=a(1-a)$.