Problem 7 - An Introduction to Mathematical Statistics and Its Applications

Question

Doomsday Airlines ("Come Take the Flight of Your Life") has two dilapidated airplanes, one with two engines, and the other with four. Each plane will land safely only if at least half of its engines are working. Each engine on each aircraft operates independently and each has probability \(p=0.4\) of failing. Assuming you wish to maximize your survival probability, which plane should you fly on?

Step-by-Step Solution

Answer

The preferred plane is chosen based on which has a higher survival probability, which is calculated by summing the probabilities of scenarios where the plane lands safely, i.e., at least half of the engines are working. This requires applying the binomial distribution and its probability mass function.

Step 1: Calculate Survival Probability for Two-Engine Plane

Firstly, consider the two-engine plane. The plane will land safely if at least one engine is working, i.e., zero or one engine fails. Let \(X\) represent the number of engines failing. \(X\) follows a binomial distribution with parameters \(n=2\) (the number of trials, which is also the number of engines in this case) and \(p=0.4\) (the probability of a trial resulting in a 'failure', which refers to an engine failing in this case). The probability that at least one engine is working is given by \(P(X \leq 1) = P(X=0) + P(X=1)\). Using the formula for the probability mass function (PMF) of a binomial distribution, \(P(X=k) = C(n, k) \cdot p^k \cdot (1-p)^{n-k}\), we can calculate \(P(X=0)\) and \(P(X=1)\) as \(P(X=0) = C(2, 0) \cdot p^0 \cdot (1-p)^2 = (1-0.4)^2 = 0.36\) and \(P(X=1) = C(2, 1) \cdot 0.4^1 \cdot (1-0.4)^{2-1} = 2 \cdot 0.4 \cdot 0.6 = 0.48\). Adding these values gives the total survival probability for the two-engine plane.

Step 2: Calculate Survival Probability for Four-Engine Plane

Next, consider the four-engine plane. This plane will land safely if at least two engines are working, i.e., zero, one, or two engines fail. Let \(Y\) represent the number of engines failing on this plane. This follows a binomial distribution with \(n=4\) and \(p=0.4\). The survival probability here is given by \(P(Y \leq 2) = P(Y=0) + P(Y=1) + P(Y=2)\). We can calculate these probabilities using the PMF of a binomial distribution. After performing these calculations, the total survival probability for the four-engine plane can be obtained.

Step 3: Choose the Plane with Higher Survival Probability

After calculating the survival probabilities for both planes, the one with the higher probability will be chosen. If the probabilities are the same, then there is no preference for either plane from a safety perspective.

Key Concepts

Probability Mass Function

The Probability Mass Function (PMF) is a fundamental concept when dealing with discrete random variables such as the number of failing engines in an airplane. It gives us the probability that a discrete random variable is exactly equal to some value.
In the case of Doomsday Airlines, we use PMF to compute the probability of varying numbers of engine failures for both the two-engine and four-engine planes.

• For the two-engine plane, which follows a binomial distribution, the PMF is calculated for zero and one engines failing, as given by the formula: \[ P(X = k) = C(n, k) \cdot p^k \cdot (1-p)^{n-k} \]where \( C(n, k) \) is the binomial coefficient.
• The survival probability is computed by summing the PMF results for the scenarios where not all engines fail.

Understanding the PMF allows decision-makers at Doomsday Airlines to determine the likelihood of different engine failure scenarios and thus assess the reliability of each airplane.

Survival Probability

Survival Probability in this context refers to the likelihood that an airplane lands safely despite some of its engines failing.
For Doomsday Airlines, it's crucial to calculate this probability in order to choose the safest flight.

• For the two-engine plane, the survival probability consists of scenarios where zero or one engine fails. It combines the probabilities calculated using the PMF to ensure at least one engine keeps running:\[ P(X \leq 1) = P(X=0) + P(X=1) \]
• For the four-engine plane, survival means that up to two engines fail, ensuring at least two are operating:\[ P(Y \leq 2) = P(Y=0) + P(Y=1) + P(Y=2) \]

By analyzing these calculations, a passenger can decide the flight with a higher survival probability, improving the chances of a safe landing.

Failure Probability

Failure Probability, on the contrary, measures the chance of an event resulting in all engines failing, causing the plane to be unable to land safely. This probability helps in understanding the risks associated with flying in either aircraft.

• For the two-engine plane, it means both engines fail, calculated as:\[ P(X=2) = C(2, 2) \cdot 0.4^2 = 0.16 \]Here, the entire failure probability is quite low at 16%.
• For the four-engine plane, the failure probability looks at the likelihood of more than two engines failing, calculated as the complement of its survival probability:\[ P(Y \geq 3) = 1 - P(Y \leq 2) \]

Understanding failure probabilities allows pilots and passengers to better gauge the risk involved with each airplane choice, particularly underlining the importance of aircraft and engine reliability in flight safety.