Ramanujan is running late to teach his lecture. He must run a mile across Cambridge to the lecture hall within ten minutes in order to get to his students on time. Beginning with attempt , he runs a distance of miles in minutes towards the lecture hall and repeats the process if he has (1) not yet arrived at the hall and (2) still has time to spare; the maximum value of is 3. The values of are chosen independently and uniformly at random between and miles, while the values of are chosen independently and uniformly at random between and minutes. If Ramanujan runs out of time during his -th attempt, then time magically stops, allowing him to finish traveling the remainder of the miles. If Ramanujan runs equal to or more than a mile total within the ten minutes, then he arrives at the hall on time. What is the probability that Ramanujan gets to the lecture hall on time?