The principle of redundancy is used when system reliability is improved through redundant or backup components. A region's government required that commercial aircraft used for flying in hazardous conditions must have two independent radios instead of one. Assume that for a typical flight, the probability of a radio failure is 0.0058. What is the probability that a particular flight will be threatened with the failure of both radios? Describe how the second independent radio increases safety in this case. What is the probability that a particular flight will be threatened with the failure of both radios? (Round to six decimal places as needed.) Describe how the second independent radio increases safety in this case. Choose the correct answer below. The second ratio makes it impossible for the airplane to crash. With one radio there is a 0.0058 probability of a serious problem, but with two independent radios, the probability of a serious problem decreases substantially. The flight becomes much safer with two independent radios. With one radio there is a 0.0058 probability of a serious problem, but with two independent radios, the probability of a serious problem increases substantially. The flight becomes much less safe with two independent radios. The second radio has no effect on safety.

When you have two events A and P with probabilities P(A)=q and P(B)=r, if the events are independent (that is to say, if the occurrence of one event does not influence the occurrence of the other), then

[tex]P(A\cap B)=P(A)P(B)[/tex]

where the event intersection means the probability of the occurrence of both events.

If we call P(A) the probability of a radio failure, as the occurrence of failure of a radio is independent of having another radio, then the probability of the failure of both radios is

(0.0058)*(0.0058) = 0.00003364

So, the probability that a particular flight will be threatened with the failure of both radios is 0.00003364.

Hence the flight becomes much safer with two independent radios