Suppose that George starts with one dollar and will gamble with it as follows: if a fair coin toss results in heads he’ll win a dollar, and if it results in tails he’ll lose a dollar; and he’ll continue to bet a dollar on subsequent flips for up to three flips, but he’ll quit as soon as he loses the first time. Letting $X$ denote George’s net profit, give the pmf of $X$ (being sure to indicate a value for every real number).
Attempted Solution
Let $H,T$ denote heads and tails, respectively. Then the sample space is
${T}$, ${HT}$, ${HHT}$, ${HHH}$
These give net profits of $-1, 0, 1$, and $3$, respectively and have probabilities of $1/2, 1/4, 1/8$, and $1/8$, respectively.
Then $$p_X(x) = \begin{cases} {\frac{1}{8}} & \text{$x = 3$} \\ \frac{1}{8} & \text{$x=1$} \\ \frac{1}{4} & \text{$x=0$} \\{\frac{1}{2}} & \text{$x = -1$}\\{0} & \text{otherwise}\end{cases}$$
Did I do this correctly?