Suppose that airplane engines operate independently in flight and fail with probability 1/5. Assuming that a plane makes a safe flight if at least one-half of its engines run, what is the probability that 4-engine plane has a successful flight?
To find the probability that a 4-engine plane has a successful flight, we need to consider the possible number of engines that could fail and still have at least half of them running.
In this case, since the plane has 4 engines, at least half of them should be running for the flight to be successful. Thus, the plane can have 4, 3, or 2 engines running.
Let's calculate the probability for each case and then add them up to get the overall probability.
1. All 4 engines are running (success):
The probability of each engine running is given as 1/5.
So, the probability of all 4 engines running is (1/5) * (1/5) * (1/5) * (1/5) = 1/625.
2. 3 out of 4 engines are running (success):
The probability of exactly 3 engines running can be calculated by selecting 3 engines out of 4 for success and 1 engine for failure.
The probability of a single engine running is 1/5, and the probability of a single engine failing is 1 - (1/5) = 4/5.
So, the probability of 3 out of 4 engines running is (4C3) * (1/5)^3 * (4/5) = 4 * (1/5)^3 * (4/5) = 64/625.
3. 2 out of 4 engines are running (success):
The probability of exactly 2 engines running can be calculated by selecting 2 engines out of 4 for success and 2 engines for failure.
The probability of a single engine running is 1/5, and the probability of a single engine failing is 1 - (1/5) = 4/5.
So, the probability of 2 out of 4 engines running is (4C2) * (1/5)^2 * (4/5)^2 = 6 * (1/5)^2 * (4/5)^2 = 96/625.
Now, we add up the probabilities from each case to get the overall probability of a successful flight:
Overall probability = (probability of all 4 engines running) + (probability of 3 out of 4 engines running) + (probability of 2 out of 4 engines running)
= 1/625 + 64/625 + 96/625
= 161/625
Therefore, the probability that a 4-engine plane has a successful flight is 161/625.