Question about the Solution for Problem 3.7b, Introduction to Probability (Bertsekas, 2nd Edition) I am currently working on the problems in Introduction to Probability (Bertsekas and Tsitsiklis, 2nd Edition) and one of the problems is as follows:

For Problem (b), the final answer is as follows:

My question:
Why do we have cumulative density for 0 <= s <= 1/t? Shouldn't the cumulative density be 0 on the 0 <= s <= 1/t interval? Based on the problem statement, my assumption is that s can only be 0 or [1/t, +inf) so anything outside of that shouldn't be possible (hence, 0 probability). Am i missing something here?
Thanks in advance.
P.S The final answer is taken from the book's Problem Solutions supplement.


my approach was to start with the values of Alvin's score (S) then map it back to the corresponding Xs (X = distance from the center) to achieve that score.

This is the right approach, with one small modification: you should map back to the corresponding values of $X$ that achieve that score or less. That is the quantity that the CDF is referring to. Remember that $F_S(s) = P(S \leq s) = \int_{-\infty}^s f(s)ds$.
For example, under the rules in (b) there is no way to get a score of $S = \frac{1}{2t}$. But there is a way to get a score of $S \leq \frac{1}{2t}$. Any dart outside the inner circle will result in $S = 0 \leq \frac{1}{2t}$.
So score values of $0 \leq s < \frac{1}{t}$ all come from missing the inner circle, which is why the CDF has a long 'flat' section over that range.
