A certain airplane has two independent alternators to provide electrical power. The probability that a given alternator will fail on a 1-hour fight is .02. What is the probability that (a) both will fail? (b) Neither will fail? (c) One or the other will fail? Show all steps carefully
Pr(one)=.02
pr(both)=.02*.02
Pr(neither)=.98*.98
pr(one or other)=.02*.98+.98*.02
To calculate the probability for each scenario, we need to use the concept of independent events.
(a) The probability that both alternators will fail is given by the product of the probabilities of each alternator failing. Thus, the probability of one alternator failing is 0.02. Since there are two independent alternators, the probability of both alternators failing is:
P(both will fail) = P(Alternator 1 fails) * P(Alternator 2 fails)
= 0.02 * 0.02
= 0.0004
Therefore, the probability that both alternators will fail is 0.0004.
(b) The probability that neither alternator will fail is the complement of the probability that at least one alternator will fail. So, we need to subtract the probability of failure from 1. The probability of one alternator failing is 0.02. Therefore, the probability that neither alternator will fail is:
P(neither will fail) = 1 - P(one or both will fail)
= 1 - [P(Alternator 1 fails) * P(Alternator 2 fails)]
= 1 - 0.0004
= 0.9996
Therefore, the probability that neither alternator will fail is 0.9996.
(c) The probability that one or the other alternator will fail can be calculated using the complementary probability. The complementary probability of one or the other alternator failing is the probability that neither alternator fails. So, the probability that one or the other alternator will fail is:
P(one or the other will fail) = 1 - P(neither will fail)
= 1 - 0.9996
= 0.0004
Therefore, the probability that one or the other alternator will fail is 0.0004.