I suggest you some exercises where my doubt comes back.
A) The rain in a certain period is represented from a random variable $T$ distributed like a $\Gamma(30,5)$. If $T=t$, the number of umbrellas $N$ sold by a certain shop complies with a Poisson law of parameter $4t$. Find $\mathbb{E}[N]$.
B) Let the random vector $\binom{X}{Y}\sim N_{2}(\binom{0}{0},\bigl(\begin{smallmatrix} 1 & \rho\\ \rho & 1\end{smallmatrix}\bigr))$ and $Z=X+Y$, find $\mathrm{Cov}(X,Z)$.
C) Mark, during his career, played $N$ tournament with $N\sim \mathrm{Geo}(s)$. Assume that Mark, in every tournament, has has a probability $p$ to win the tournament, independently. Let $T$ the number of tournament wins. Find $\mathbb{E}[T]$.
Well. I know that $\mathbb{E}[X]=\mathbb{E}\bigl[\mathbb{E}[X \mid Y] \bigr] =\mathbb{E}[Y]$, but:
-for A) we have $\mathbb{E}[N]=\mathbb{E}\bigl[\mathbb{E}[N \mid T] \bigr] =4\mathbb{E}[T]= \ldots$
-for B) we have $\mathrm{Cov}(X,Z)= \ldots =\mathbb{E}[X^2Y]=\mathbb{E}[X^2]\mathbb{E}\bigl[\mathbb{E}[Y \mid X] \bigr] =\mathbb{E}[X^2]\rho\mathbb{E}[X]= \ldots$
-for C) we have $\mathbb{E}[T]=\mathbb{E}\bigl[\mathbb{E}[T \mid N] \bigr] =p\mathbb{E}[N]= \ldots$
Why, formally, do we have for A) $4$, for B) $\rho$ and for C) $p$?
Thanks in advance for any clarification!
A) Observe that $\mathbb{E}\left[N\mid T=t\right]=\color{red}4t$ justifying $\mathbb{E}\left[N\mid T\right]=\color{red}4T$.
B) $\mathsf{Cov}\left(X,Z\right)=\mathsf{Cov}\left(X,X+Y\right)=\mathsf{Cov}\left(X,X\right)+\mathsf{Cov}\left(X,Y\right)=\mathsf{Var}X+\mathsf{Cov}\left(X,Y\right)=1+\rho$
C) Observe that $\mathbb{E}\left[T\mid N=n\right]=n\color{red}p$ justifying $\mathbb{E}\left[T\mid N\right]=N\color{red}p$.
I don't see your logic in the way you try to handle case B) and further hope that A) and C) are enough to understand the presence of the factors $\color{red}4$ and $\color{red}p$ respectively.