It depends on the trial, and is normally denoted by p.
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What is the symbol for a Probability of success in a binomial trial?
No, in general is not. It is only symmetric if the probability of success in each trial is 0.5
The requirements are that there are repeated trials of the same experiment, that each trial is independent and that the probability of success remains the same.
[(1 - p)/(1 - pet)]r for t < -ln(p) where p = probability of success in each trial, r = number of failures before success.
For any particular trial, the total probability is 1.