How to calculate probability of touching a take-profit without touching a stop-loss (no-dividend stock, infinite time)?
a) you can run a Monte Carlo simulation in which you model stock price movements and then you can look at the future pay off as a function of path dependency into which you incorporate your stop losses and take profits. Done this over many iterations you will be able to derive your probabilities. Caveat here is your result will be strongly dependent on your model assumption of how you derive stock price movements.
b) you could run back tests over actual pricing data. You generate trading signals, associated stops and take-profit targets and you derive your probabilities through simple counting processes of how often your stop loss was hit vs take profit targets.
I would strongly prefer the latter approach, this is by the way a very common way to also calculate the quality of entry and exit signals (not using stop loss and target levels but by keeping track of how prices performed after the entry/after the exit).
P.S. I find the question correctly worded and I think it actually makes sense.