Probability a draw-until-threshold score ends at or below a limit

A player starts with `0` points and draws repeatedly. Each draw adds an integer chosen uniformly at random from `1` to `max_points` (inclusive) to the player's total, independently of all other draws. The player keeps drawing while the total is strictly less than `threshold` and stops as soon as the total reaches `threshold` or more. Return the probability that the final total is at most `limit`. ### Function Signature ```python def probability_within_limit(limit: int, threshold: int, max_points: int) -> float: ``` ### Rules - If `threshold` is `0`, the player never draws and the final total is `0`. - Return the probability as a float. An answer is accepted if it is within `1e-5` (absolute) of the exact probability. ### Constraints - `0 <= threshold <= limit <= 10^4` - `1 <= max_points <= 10^4` ### Examples **Example 1** ```text Input: limit = 6, threshold = 1, max_points = 10 Output: 0.6 ``` The player draws exactly once and stops. The total is uniform on 1 to 10, and 6 of those 10 values are at most 6. **Example 2** ```text Input: limit = 2, threshold = 2, max_points = 2 Output: 0.75 ``` With probability 0.5 the first draw is 2 and the player stops at 2. Otherwise the total is 1 and the player draws again, ending at 2 or 3 with probability 0.5 each. The final total is at most 2 with probability 0.5 + 0.25 = 0.75. **Example 3** ```text Input: limit = 4, threshold = 3, max_points = 3 Output: 0.85185 ``` The final total is 3, 4 or 5 with probabilities 16/27, 7/27 and 4/27, so the answer is 23/27, about 0.85185.