1. The problem is to explain why the formula $$P(success) = P(S) \times P(success \mid S) + P(\neg S) \times P(success \mid \neg S)$$ holds. 2. This formula is an application of the Law of Total Probability. It breaks down the probability of success into two mutually exclusive cases: when event $S$ occurs and when event $\neg S$ (not $S$) occurs. 3. The term $P(S)$ is the probability that event $S$ happens. 4. The term $P(success \mid S)$ is the conditional probability of success given that $S$ has occurred. 5. Similarly, $P(\neg S)$ is the probability that event $S$ does not happen. 6. And $P(success \mid \neg S)$ is the conditional probability of success given that $S$ has not occurred. 7. Since $S$ and $\neg S$ are complementary and cover all possible outcomes, the total probability of success is the sum of the probabilities of success in each case weighted by the probability of that case. 8. Therefore, the formula sums the weighted probabilities of success conditioned on $S$ and $\neg S$ to get the overall probability of success. Final answer: $$P(success) = P(S) \times P(success \mid S) + P(\neg S) \times P(success \mid \neg S)$$