values: which is close to the sum of the two probabilities. The example shows that each critical point of failure increases the probability of system failure by approximately that item's probability of failure. If there are two completely independent scuba sets at the diver's disposal, either one of which is sufficient to allow the diver a safe return, then both sets must fail during the same dive to cause a fatal outcome. These items work in parallel - all must fail for the system to fail. The probability of this happening is extremely low for reliable equipment. Assuming independence of failure events, each duplicate redundant item added to the system decreases the probability of system failure according to the formula:- where: – number of components – probability of component i failing – the probability of all components failing (system failure) Taking two independent sets with the same probability of failure calculated in the example above: , and Therefore: Substituting values: It is clear from the example that redundancy reduces the risk of system failure very rapidly, and conversely, that disregarding a failure of a redundant item increases the probability of system failure equally rapidly. Closed circuit Open circuit scuba has a small number of fairly rugged and reliable components, each with a small number of failure modes and a low probability of failure. Most of these components remain present in closed circuit scuba, but there are also a number of additional items which could fail. Therefore, the rebreather architecture is inherently more likely to fail, and it is necessary to provide redundancy of critical components to provide reliability even approaching that of open circuit scuba. It is also more important to provide full redundancy of breathing gas supply as some rebreather failure modes do not allow safe ascent. Bailout to open circuit is the simplest and most robust option, but for dives where a long return under an overhead, or long decompression are necessary, open circuit can be impractically bulky. There is a point at which closed circuit bailout becomes a more manageable option, and the requirement for ability to return safely from any point on the planned dive profile makes it necessary for the breathing loop and gas supplies to be fully independent, though the ability to make use of the primary gas supply in the bailout rebreather can considerably extend the range for a small added complexity, using