Would appreciate if you could check if I answered the questions correctly:
$X$ is a Negative Binomial random variable with the parameters $\frac{1}{2}$ and $r =1,2,3,\ldots$.
$Y$ is a random variable that is defined as: $Y = \begin{cases} X, & x \le r+1 \\ X-1, & x \ge r+2 \end{cases}$
(1) Find $\Pr\{ Y = r+1 \}$
The result I got: $r\,(\frac{1}{2})^{r+2}$. Since we need, according to the definition of Negative Binomial, to calculate the probability of trials to make $r$ success, then we can calculate it by multiplying the parameters ($1/2$ and $r$) times $r+1+1$ to cover it all.
(2) Find $\mathbb{E}[Y]$
The result I obtained is $2r-1-\frac{r+2}{2^r}$. I calculated it using $r/p$ while trying to cover all of $Y$.
Hope I did this one correctly. Would appreciate your comments or corrections.
Your description in (1) and the fact that you quoted $\mathbb{E}[X] = r/p$ for (2), I take it that you adopt the total number of trials for the definition of $X$ the Negative Binomial. \begin{align*} \Pr\bigl\{ Y = r+1 \bigr\} &= \overbrace{\Pr\bigl\{X = r+1 \bigr\} }^{Y=X} + \overbrace{ \Pr\bigl\{ X = r+2\bigr\} }^{Y=X-1} \qquad\text{, by definition of $Y$} \\ &= {r\choose r-1} p^rq + {r+1\choose r-1} p^r q^2 \Bigg|_{p=q=1/2}\\ &= r \cdot \frac1{ 2^{r+1}} + \frac{(r+1)r}{2!} \cdot \frac1{ 2^{r+2}} \\ &= \frac{ 4r }{ 2^{r+3}} + \frac{ r^2 + r }{ 2^{r+3}}\\ &= \frac{ r(r+5) }{ 2^{r+3}} \end{align*} Similarly, the expectation splits into two parts. \begin{align*} \mathbb{E}[ Y ] &= \Bigl\{ \text{contribution from $X \leq r+1$} \Bigr\} + \Bigl\{ \text{from $X \geq r+2$} \Bigr\} \\ &= \sum_{k=r}^{r+1} \Bigl[ \color{magenta}k \cdot \Pr\bigl\{ X = k \bigr\} \Bigr] + \Biggl\{ \mathbb{E}[ X-1 ] - \sum_{k=r}^{r+1} \Bigg[ (\color{magenta}k-1) \cdot \Pr\bigl\{ X = k \bigr\} \Bigg] \Biggr\} \\ \end{align*} After canceling the $\color{magenta}{\text{magenta }k}\,$, we have \begin{align*} \mathbb{E}[ Y ] &= \sum_{k=r}^{r+1} \Bigl[ 1 \cdot \Pr\bigl\{ X = k \bigr\} \Bigr] + \mathbb{E}[ X-1 ]\\ &= \frac1{ 2^r} + \frac{ r }{ 2^{r+1}} + \frac{r}p\Bigg|_{p = 1/2} - 1 \\ &= 2r-1 + \frac{ r+2 }{ 2^{r+1}} \end{align*} Your own answer is just off by a minus sign.