A selection has three rounds. A candidate clears them independently with probabilities 0.8, 0.75 and 0.5. What is the probability that the candidate clears all three rounds?

Independent stages multiply: $0.8\times 0.75\times 0.5=0.6\times 0.5=0.3$. Adding the three probabilities gives 2.05, a number greater than 1, so it cannot be a probability at all — a useful check that the wrong operation has been used. The value 0.6 stops after the first two rounds, and 0.7 is the complement 1 − 0.3, the probability of failing at least one round. Notice how quickly a chain of stages eats into the chances: each round on its own looks comfortable, yet fewer than a third of the candidates get through all three.