Suppose the probability of success of an event is `s` and the probability of failure is `t.` What is `s+ t` ?
A sample space is the set of all possible outcomes of experiments. Any event may consist of zero, one or more outcomes.
The first thing we need to know to answer this question is that, by definition, the probability of entire sample space is 1 (or 100%, if you want).
Also we need to know that for any two incompatible events (those that have no common outcomes) A and B, the probability of event "A or B" is the sum of their probabilities, P(A or B) = P(A) + P(B).
Now we can solve the problem. Denote our event as A and its failure (not A) as B. Then it is given that P(A) = s and P(B) = t. Also A and B are incompatible and any outcome is in A or in B (success or failure, no third possibility), so (A or B) is an entire sample space.
Finally, we have
1 = P(entire sample space) = P(A or B) = P(A) + P(B) = s + t.
So the answer is s + t = 1.