observing from in the water at the surface, and is ready and able to dive to the rescue if the diver loses consciousness during the ascent. This recommendation has an obvious point of failure when the visibility does not allow the buddy to observe the diver throughout the dive. Scuba diving Diving using self-contained underwater breathing apparatus was developed after surface supplied diving, and was intended as a method of improving the mobility and horizontal range of the diver who is not restricted by a physical connection to a surface gas supply. The diver has a larger gas supply than the freediver, and this allows a greatly extended underwater endurance, and lower risk of drowning, but at the cost of higher risk from decompression sickness, lung over-pressure barotrauma, nitrogen narcosis, oxygen toxicity and hypothermia, all of which must be limited by procedural and engineering controls, and personal protective equipment. For acceptable safety the diver must be able to survive any reasonably foreseeable single point of failure. For scuba equipment this implies that the failure of any single item of equipment should not put the diver out of reach of a breathing gas supply. Open circuit In the case of a single cylinder scuba set with a single first stage, and a single second stage, each of these items has a low but non-zero probability of failure. The components work in series - if any one of them fails, the system fails. It is equivalent to a single chain in which if any link fails, the chain breaks. When the dive is very shallow, the diver can safely escape to the surface, and when there is another diver right there with spare gas at the time of failure, they can share gas. At other times, a failure of a single item can kill the diver. Assuming independence of failure events, each item that can cause failure of the combined system is a critical point of failure and increases the probability of system failure. For the system not to fail, all items must not fail according to the formula: where: – number of components – probability of component i failing – the probability of all components failing (system failure) As a purely illustrative example, if there is a 1 in 100 probability of a regulator failure, and a 1 in 1000 probability of a scuba cylinder failure then , and Therefore: Substituting