How to show $U$ and $V$ are not independent random variable?

Question

$U$ stands for the number of trials to get the first head, $V$ stands for the number of trials to get two heads.

I used hand-waving proof, saying that you could not have the two heads trials without having the first head occur, but I don't know how to mathematically prove that the two random variables must be dependent.

Answer 1

The variables are independent if $P(U=u, V=v)=P(U=u)\cdot P(V=v)$ for all values $u,v$.

Therefore, to prove that the variables are not independent, all you need to do is find one pair $u,v$ such that

$P(U=u, V=v)\neq P(U=u)\cdot P(V=v)$

To do that, you can go one of two ways:

Option $1$: Try a little brute force search. Plug in a couple values $u,v$ and see if the equation holds.

Option $2$: Since the right side of the equation will always be nonzero, think about how you could make the left side zero. Can you think of some pair of numbers $u,v$ such that it's impossible to get two heads in $v$ trials, but only one head in $u$ trials?