Question: The probability that two or more of a system's 10 components fail between consecutive inspections is 0.005, while the probability that only one component fails is 0.1. When two or more components fail, a re-evaluation of the system is initiated during which all failed components, and those deemed unreliable, are replaced. Otherwise, components are replaced upon their failure. Find the probability that a system re-evaluation occurs before any component is individually replaced. (Hint. Let B = {system re-evaluation occurs}, C = {a component is individually replaced}, and consider a new experiment with reduced sample space A = B ∪ C. The desired probability is the probability of B in this new experiment. See also Example).

Example: Fair game with an unfair coin. Suppose that when a biased coin is flipped it results in heads with probability p. A fair game with such an unfair coin can be played as follows: Flip the coin twice. If the outcome is (H, H) or (T, T) ignore the outcome and flip the coin two more times. Repeat until the outcome of the two flips is either (H, T) or (T, H). Code the first of these outcomes as 1 and the second as 0. Prove that the probability of getting a 1 equals 0.5.