A certain airplane has two independent alternators to provide electrical power. The probability that a given alternator will fail on a 1- hour flight 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.
(a) (0.02)^2 = 0.0004
(b) (0.98)^2 = 0.9604
(c) (0.02)(0.98) + (0.98)(0.02) = 0.0392
Note that the sum is 1.0000
To calculate the probabilities in this scenario, we will use the concept of independent events. We'll start by solving each part individually:
(a) Probability that both alternators will fail:
For both alternators to fail, we need the first alternator to fail AND the second alternator to fail. Since these are independent events, we can multiply their individual probabilities:
P(Both will fail) = P(First will fail) * P(Second will fail) = 0.02 * 0.02 = 0.0004
(b) Probability that neither alternator will fail:
The probability that the first alternator will not fail is the complement of it failing, which is 1 - P(First will fail) = 1 - 0.02 = 0.98. Similarly, the probability that the second alternator will not fail is also 0.98. Since these events are independent, we multiply their individual probabilities:
P(Neither will fail) = P(First will not fail) * P(Second will not fail) = 0.98 * 0.98 = 0.9604
(c) Probability that one or the other alternator will fail:
To calculate this probability, we need to consider two scenarios: either the first alternator fails while the second one doesn't, or the second one fails while the first one doesn't. These scenarios are mutually exclusive, so we can add their probabilities together:
P(One or the other will fail) = P(First fails and Second doesn't) + P(Second fails and First doesn't)
For the first scenario, the probability is:
P(First fails and Second doesn't) = P(First will fail) * P(Second will not fail) = 0.02 * 0.98 = 0.0196
Similarly, for the second scenario:
P(Second fails and First doesn't) = P(First will not fail) * P(Second will fail) = 0.98 * 0.02 = 0.0196
Finally, we add these two probabilities together:
P(One or the other will fail) = 0.0196 + 0.0196 = 0.0392
Therefore, the probabilities are:
(a) P(Both will fail) = 0.0004
(b) P(Neither will fail) = 0.9604
(c) P(One or the other will fail) = 0.0392