The reliability of a deep-sea submersible is monitored via three independent systems. Let S S\,S be the event that the hull integrity sensor triggers an alarm, P P\,P be the event that the power supply stability module reports a fault, and C C\,C be the event that the communication array fails.
Historical data indicates that P(S)=0.35P(S) = 0.35P(S)=0.35, P(P)=0.45P(P) = 0.45P(P)=0.45, and P(S∩P)=0.14P(S \cap P) = 0.14P(S∩P)=0.14.
Determine the probability that the submersible experiences either a hull alarm or a power fault, P(S∪P)P(S \cup P)P(S∪P).
Calculate P(P′∣S′)P(P' | S')P(P′∣S′), the probability that the power supply is stable given that no hull alarm was triggered.
The probability of a communication failure is P(C)=0.20P(C) = 0.20P(C)=0.20.
It is known that the hull alarm and communication failure are mutually exclusive events, and the power supply module and communication array operate independently.
Find P(P∩C)P(P \cap C)P(P∩C).
Describe the configuration of a Venn diagram for events SSS, PPP, and CCC, and state the probabilities for each of the six distinct regions.
Find the probability that neither a power fault nor a communication failure occurs, P((P∪C)′)P((P \cup C)')P((P∪C)′).
