Cumulative probabilities/normal distribution/series of trials

[BvlyHls90210]

Assume a normal distribution. I understand how to calculate the probability of something being X standard deviations above the mean, for example, mean is 10, standard deviation is 5, probability of >15 is (1-.67)/2.

Furthermore, I understand that if we have say 100 trails, the new mean should be 100*the old mean, new variance is 100X the old variance (square result to get st dev). And it is similarly straightforward to determine the probability of being X st devs above the mean (works just like the above example).

But here is my question, how do I determine the probability of the result being > x st devs above the mean at any point in the 100 trial run, not just at the end?

I mean I could calculate the probability for 1 run, 2 runs, 3 runs, etc up to 100, but I see no way to derive the cumulative probability that it happens at any point during that run.

Any ideas?

Let me know if you need any clarification of my question.