There's some good analysis on the math itt, but I went a bit further and built a random walk simulator to mimic this contest with some additional strategy. The problem with using straight-line probability assumptions is that it implied that Timex knows exactly when to start a winning run or end a losing one (i.e. he knows to start a 90/100 run on the 1st shot or ends losing runs only when he's drawing dead). In reality, there is likely a major advantage or disadvantage to be found in Mike's "reset" strategy.
I tested several strategies just for the sake of argument. Base assumptions (I am in no way saying this is the optimal strategy. I just wanted to try something):
1) Mike resets if he misses the first shot.
2) Past the first shot, Mike resets anytime his remaining FT% needed to clear exceeds 93% (i.e. the floor would be that Mike makes the 1st and misses the next 4, then he'd need to finish 89/95 = 93.6%, so he resets after the fifth shot).
3) The 93% rule is aborted if Mike is within 20 shots of winning. So if Mike is 70/80 and needed to finish 20/20, the 93% reset rule is waived and he keeps going until a miss. I call this the YOLO rule because he's surely wasting plenty of shots but he probably won't make it this far very often, so YOLO.
I tested every free throw percentage from 70% - 90%, 20,000 times each. Here are the win percentages and shots per winning attempt (you'll note that there were no winning simulations at 74% shooting or lower. Not saying that's impossible. Just saying that it didn't happen in the first 20,000 trials):
But I then took this even further and tinkered with rule #1. I tested each FT% where the reset comes after having to make up to the first 10 shots. In other words, if I set the threshold to 5, then he immediately resets if he misses any of the first 5. If the threshold is 10, then he has to make the first 10 to continue. If he makes it out of the initial period only then does he proceed with the other rules. Just to quickly visualize what the results showed:
At a 5 shot reset rule, there are now no winning trials below a 77% shooter. However, the shots per win for a 77% shooter went from 111k in the base case, 96k with the 5 shot reset, all the way down to 59k at a 10 shot reset. Also note that the average shots per sequence (SPS) predictably also goes way down. The 77% shooter averages 14.8 SPS in the base rules, 9.6 SPS with the 5 shot reset, and 6.0 at the 10 shot reset. His win percentage per sequence will go way down but he'd churn losing sequences at a much faster rate.
I could probably go further with testing other ideas, but I've certainly come away with a greater sense that the math supports Timex's side. Of course this is way easier to take millions of shots on a computer than it is in practice. The question really is how quickly he can get up to being a regular 77% shooter. If he can do that by say October, then I think he's way live to beat this by the end of year. But I have absolutely no idea how feasible that is or isn't