Expected values of two random variables related to a simple urn problem

Question

In an urn there are $u$ balls, $b$ of which are black.

If we perform $n$ trials of one ball at a time with replacement, the probability of the event $E$ to get $n$ times a black ball is $P(E)=\left(\frac{b}{u}\right)^n$, whereas the probability of the event $L$ to get at least one black ball is $P(L)=1-\left(\frac{u-b}{u}\right)^n$.

Let $X$ be the non-negative, integer-valued random variable representing the number of trials it takes to get a success for the event $E$, and $Y$ the non-negative, integer-valued random variable representing the number of trials it takes to get a success for the event $L$.

What are the expected values $\mathbb{E}[X]$ and $\mathbb{E}[Y]$?

My attempt for $\mathbb{E}[X]$:

I denote with $P(E;k)$ the probability to get a success for the event $E$ at the trial $k$.

The variable $X$ can assume only the value $n$, therefore, by definition it should be $$\mathbb{E}[X]=\sum_{k=1}^n k P(E;k)=\sum_{k=1}^n k P(E)\delta_{n,k}=nP(E).$$

My attempt for $\mathbb{E}[Y]$:

I denote with $P(L;k)$ the probability to get a success for the event $L$ at the trial $k$.

The variable $Y$ can assume any value from $1$ to $n$, therefore, by definition, it should be $$\mathbb{E}[Y]=\sum_{k=1}^n k P(L;k)=\sum_{k=1}^n k \left[1-\left(\frac{u-b}{u}\right)^k\right].$$

Are these calculations correct?

Answer 1

For the alternative approach in the comments, the Pascal distribution applies: Repeatedly draw a ball with replacement from an urn with $u$ balls, $b$ of which are black, stop when you have $n$ black balls. Let $X$ be the number of balls drawn. The probability distribution of $X$ is $$P(X=N)=\binom{N-1}{n-1} (b/u)^{n}(1-b/u)^{N-n},\;\;N\geq n,$$ with expectation value $\mathbb{E}(X)=nu/b$.

Answer 2

In your definitions, you have confused:

$$P(E;k)$$

This actually means:

$$P(X=k|E)$$

(which equals $\delta_{kn}$ and gives the correct value that $\mathbb E(X)=n$).

and not, as you have used:

$$P(E|X=k)$$

which equals $P(E)\delta_{kn}$.

See Conditional Probability for more information.