The following question is from Blitztein-Hwang, Introduction to Probability:
Independent Bernoulli trials are performed, with probability 1/2 of success, until there has been at least one success. Find the PMF of the number of trials performed.
Let $X$ be the number of trials performed.
When I first attempted the question, I concluded that $P(X=k) = 1-(\frac{1}{2})^k$ via the principle of inclusion-exclusion and the independence of events. My thought process here was the following: because the question says until there has been at least one success, then for $k$ trials, we need at least one success, i.e., we can get up to $k$ success.
Then, after a quick search, I stumbled on this previous question: Bernoulli trials with at least 1 success and 1 failure. In the answer, it is stated that $X$ has a geometric distribution, i.e., $P(X=k) = (1-\frac{1}{2})^{k-1}\cdot \frac{1}{2} = (\frac{1}{2})^k$, but this assumes that after $k$ trials, we only have one success, so it seems more close to until there has been a success and not until there has been at least one success
Please, can someone help me understand which case better reflects the question and why?
Your error is in this line:
The key detail is that you're counting until the first success. This doesn't mean you can have up to $k$ successes; it means instead that your first $k-1$ trials could not have had a success, but that your $k^{\text{th}}$ one did, making it the first of them. That's the role of the word until in this context; it implies that we stop when we find a success. (Yes, the use of "at least one" becomes necessarily strange in this formulation.)