An aircraft is equipped with three engines that operate independently. The probability of an engine failure is .01. What is the probability of a successful flight if only one engine is needed for the successful operation of the aircraft?

To find the probability of a successful flight, we need to consider the probabilities of all possible scenarios. In this case, we have three engines, and only one is needed for a successful flight.

Let's break down the different scenarios:

Scenario 1: All three engines are functioning correctly.
Scenario 2: Two engines are functioning correctly, but one fails.
Scenario 3: One engine is functioning correctly, but two fail.
Scenario 4: All three engines fail.

We know that each engine's failure probability is 0.01, so the probability of an engine functioning correctly is 1 - 0.01 = 0.99.

Now, let's calculate the probabilities for each scenario:

Scenario 1:
Since all three engines are working, the probability of this scenario is given by:
P(All three engines working) = P(Engine 1 working) * P(Engine 2 working) * P(Engine 3 working)
= 0.99 * 0.99 * 0.99
= 0.99^3

Scenario 2:
In this case, two engines are functioning correctly, and one fails. We have three choices for the failed engine (Engine 1, Engine 2, or Engine 3). The probability of this scenario is given by:
P(Two engines working, one fails) = 3 * P(Engine working) * P(Engine working) * P(Engine fails)
= 3 * 0.99 * 0.99 * 0.01
= 3 * 0.99^2 * 0.01

Scenario 3:
In this case, one engine is functioning correctly, and two engines fail. Again, we have three choices for the functioning engine. The probability is given by:
P(One engine working, two fail) = 3 * P(Engine working) * P(Engine fails) * P(Engine fails)
= 3 * 0.99 * 0.01 * 0.01
= 3 * 0.99 * 0.01^2

Scenario 4:
In this case, all three engines fail. The probability is given by:
P(All three engines fail) = P(Engine fails) * P(Engine fails) * P(Engine fails)
= 0.01 * 0.01 * 0.01
= 0.01^3

Now, to find the probability of a successful flight, we need to add up the probabilities of the scenarios where the flight is successful, which is Scenario 1, Scenario 2, and Scenario 3.

P(Successful flight) = P(All three engines working) + P(Two engines working, one fails) + P(One engine working, two fail)
= 0.99^3 + 3 * 0.99^2 * 0.01 + 3 * 0.99 * 0.01^2

Please do the calculations to get the final answer.