I have to calculate probability for m or more consecutive outcomes in a set of trials. The number of trials (N) is very large. There are only two outcomes O1 and O2. Probability p of O1 is 6204 in 6205 and probability of O2 is (1-p). Hence, probability of O1 is very large as compared to that of O2. The trials are independent.
I have to calculate the probability of (m or more) consecutive O1 outcomes.
I looked online and the formula/calculations are pretty complicated.
But I do not need the exact probability. If I can get bulk of it, will do for me. All I have to do, is to establish a minimum probability keeping the calculations simple.
So, I thought of adding following for my purpose -
First m outcomes can be all O1 followed by O2, with probability = $p^m$(1-p).
First m+1 outcomes can be all O1 followed by O2, with probability = $p^1p^m$(1-p).
First m+2 outcomes can be all O1 followed by O2, with probability = $p^2p^m$(1-p).
and so on ... and add them in the end.
If I continue this way, after some iterations, the values become so insignificantly small that it does not matter whether I continue further or stop there. But by that time, I have already achieved the minimum value I need to establish.
Question is - Is there any flaw in this logic for establishing a lower limit of probability for (m or more) consecutive O1s. If so, what is the flaw and what is its fix.
With N being large enough, would it be ok to apply same logic to last m, m+1, m+2 ... to double my lower limit?