Dependence of coin tosses

Question

We are tossing a coin $m$ times

The probability of heads $\Bbb P(H)$ is anywhere in $(0,1)$. It doesn't have to be a fair coin basically.

Random variables $X$ and $Y$ are defined as the total numbers of heads and tails in $m$ tosses.

I am trying to show $X$ and $Y$ are dependent using $\Bbb P(XY)=\Bbb P(X)\Bbb P(Y)$, but I got stuck on defining $\Bbb P(XY)$.

Answer 1

You just need to show that $\mathbb P(X=x,Y=y)\neq \mathbb P(X=x)\mathbb P(Y=y)$ for some $x$ and $y$.

We have that

$$\mathbb P(X=m, Y=0)=\mathbb P(X=m)\neq \mathbb P(X=m)\mathbb P(Y=0)$$