As you recall from the introduction, the probability that a given trial will produce a certain outcome, often called "success", is indicated by a real number p, where 0 ≤ p ≤ 1. If p is 0 the trial never succeeds; if p is 1 the trial always succeeds. If p is a/b (rational), the experiment is indistinguishable from a black box that selects one of b choices at random, such that a of these choices lead to success. Real values of p are defined by continuity.
If two trials are independent, and they succeed with probabilities p1 and p2 respectively, the probability that both trials will succeed is p1×p2. This is easy to demonstrate when p1 and p2 are rational. The ratio of successful outcomes to total outcomes is a1×a2 over b1×b2. This is a consequence of the fundamental theorem of combinatorics. By continuity, the composite probability is p1×p2 for all real numbers p1 and p2.
The probability of failure is often denoted q1, where q1 = 1-p1. Both trials will fail with probability q1×q2. At least one of the two trials succeeds with probability 1-q1q2.