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 $Bbb P(XY)$.

Remy · Accepted Answer

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

Dependence of coin tosses

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