I cant figure this out, please explain.
A certain airplane has (2) independent alternators to provide electrical power. The probablility that a given alternator will fail on a 1 hr flight is .02.
What is the probability that (a) both will fail (b) neither will fail (c) one or the other will fail? Show steps...
The probability that both will fail is 0.02 x 0.02. The probability that neither will fail is (1-0.02)*(1-0.02) = 0.98 x 0.98. The probability that one or the other will fail is (2 x 0.02 x 0.98). Why? Because there are two situations to consider here: Alternator A fails but alternator B doesn't, and alternator B fails but alternator A doesn't. The probability of each of them is the same, i.e. 0.02 x 0.98. Check that the sum of the probabilities of all three scenarios adds up to 1, because they ought to.
I used your example for my other problem and I was able to figure it out. Thank You!!!!
To calculate the probability, we will use the concept of independent events.
Let's break down the problem into each subpart:
(a) Probability that both alternators will fail:
Since both alternators are independent, the probability that both will fail is given by the product of their individual probabilities of failure.
P(both will fail) = P(first alternator fails) * P(second alternator fails)
P(both will fail) = 0.02 * 0.02
P(both will fail) = 0.0004
(b) Probability that neither alternator will fail:
The probability of neither alternator failing is equal to the complement of both alternators failing. So,
P(neither will fail) = 1 - P(both will fail)
P(neither will fail) = 1 - 0.0004
P(neither will fail) = 0.9996
(c) Probability that either one will fail:
To calculate the probability that either one of the alternators will fail, we can use the principle of complement and subtract the probability of both alternators not failing from 1.
P(one or the other will fail) = 1 - P(neither will fail)
P(one or the other will fail) = 1 - 0.9996
P(one or the other will fail) = 0.0004
So, the probabilities are:
(a) Probability that both will fail: 0.0004
(b) Probability that neither will fail: 0.9996
(c) Probability that one or the other will fail: 0.0004
To find the probability of events, we can use the concept of independent probabilities. Let's go through each part step by step.
(a) Probability that both alternators will fail:
Since the alternators are independent, their failures are also independent events. We can use the multiplication rule for independent events:
P(both will fail) = P(first alternator will fail) * P(second alternator will fail)
Given that the probability that a given alternator will fail is 0.02, we can substitute these values into the equation:
P(both will fail) = 0.02 * 0.02
P(both will fail) = 0.0004
So, the probability that both alternators will fail on a 1-hour flight is 0.0004.
(b) Probability that neither alternator will fail:
The probability that an alternator will not fail on a 1-hour flight is simply the complement of it failing. In other words, it is 1 minus the probability of failure.
P(neither will fail) = 1 - P(either one will fail)
Since there are two alternators, we need to consider the probability of either one failing. We can use the addition rule for mutually exclusive events:
P(either one will fail) = P(first alternator will fail) + P(second alternator will fail)
Using the given probability that a given alternator will fail (0.02), we can substitute these values into the equation:
P(either one will fail) = 0.02 + 0.02
P(either one will fail) = 0.04
Now we can calculate the probability that neither alternator will fail:
P(neither will fail) = 1 - P(either one will fail)
P(neither will fail) = 1 - 0.04
P(neither will fail) = 0.96
So, the probability that neither alternator will fail on a 1-hour flight is 0.96.
(c) Probability that one or the other will fail:
To find the probability that one or the other alternator will fail, we need to consider two scenarios: either the first fails while the second doesn't or the first doesn't fail while the second does. Again, we can use the addition rule for mutually exclusive events:
P(one or the other will fail) = P(first fails and second doesn't fail) + P(first doesn't fail and second fails)
Using the given probability that a given alternator will fail (0.02), we can substitute these values into the equation:
P(one or the other will fail) = P(first alternator will fail) * P(second alternator will not fail) + P(first alternator will not fail) * P(second alternator will fail)
Since the alternators are independent, we can use the multiplication rule for independent events:
P(one or the other will fail) = 0.02 * (1 - 0.02) + (1 - 0.02) * 0.02
P(one or the other will fail) = 0.02 * 0.98 + 0.98 * 0.02
P(one or the other will fail) = 0.0196 + 0.0196
P(one or the other will fail) = 0.0392
So, the probability that one or the other alternator will fail on a 1-hour flight is 0.0392.