Aboiler has five identical relief valves. the probability that any particular valve will open on demand is .95. assuming independent operation of the valves, calculate p(at least one valve opens) and p(at least one valve fails to open).

To calculate P(at least one valve opens), we can find 1 - p(no valves open).
This is equal to 1 - (0.05)^5, or 0.9999996875. 
P(at least one valve fails to open) is equal to 1 - p(all valves open and none fail). This is 1 - (0.95)^5, or 0.2262190625.