Probability this game ends in k darts thrown Marissa is playing a game where she throws darts at a dartboard until her most recent dart is further away from the bullseye than any dart she threw previously. What is the probability that this game ends in $k$ darts thrown?
My solution:
The probability is essentially equal to how many valid orderings of the darts there are over $k!$. Thus, label each dart from $1$ to $i$,corresponding to when it was thrown. Similarly, label the positions, according to how close they are to the bullseye(with $1$ being the position closest to the bullseye). We will call the first dart and first position $d_1$ and $p_1$ respectively. Now, we know that $d_k$ needs to be assigned to $p_k$, as the game ends at $k$ darts. Thus, we need to order the first $k-1$ darts. It's clear that $d_1$ must be assigned to $p_{k-1}$, as otherwise, there will be a time where the most recent dart thrown will be the furthest from the bullseye, ending the game before $k$. But after we assign $d_1$ to $p_{k-1}$, the rest of the ordering can be any which way, right? So it's $\frac{(k-2)!}{k!}$?
However, the answer is actually $\frac{k-1}{k!}$, and I fail to see how.

Your reasoning is correct, and your answer, $\dfrac{(k-2)!}{k!}=\dfrac{1}{k(k-1)}$, is correct.
The supposed answer, $\dfrac{k-1}{k!}$, is wrong. The usual way to show an answer to a question like this is wrong, is to show that $\sum_{k=2}^\infty {\dfrac{k-1}{k!}}\neq1$, but in this case the sum equals $1$ by coincidence. So instead, let's consider the probability that the game ends in $3$ darts thrown. This means the darts, in order of increasing distance from the bullseye, were: second dart, first dart, third dart. The probability of this happening is $\dfrac{1}{3!}=\dfrac16$, which contradicts the supposed answer $\dfrac{k-1}{k!}=\dfrac13$.
By the way, another way to phrase the question is:

Marissa is playing a game where she throws darts at a dartboard until her most recent dart is further away from the first dart she threw. What is the probability that this game ends in $k$ darts thrown?

