91Ó°ÊÓ

Probability calculation is the process of determining the likelihood of a specific outcome. In this context, we are interested in the probability that a certain number of airplane engines will work or fail. Probability is expressed as a number between 0 and 1, where 0 means something will not happen, and 1 means it definitely will happen.

To begin calculating probabilities, it's essential to recognize the basic principle:

• The probability of all possible outcomes in a scenario adds up to 1.
• The probability of a single outcome is the ratio of the number of successful outcomes to the total number of possible outcomes.

When dealing with multiple independent events, such as the functioning of airplane engines, each event contributes to the overall probability. By understanding the context (e.g., the likelihood of an engine working), we can calculate the overall probability of success based on a sequence of at least two events succeeding.

Specifically, we calculate the probability of failure and success using tools like combinations and permutations to account for various scenarios.

When discussing probability, independent events are a crucial concept. Independent events are those that do not influence one another. In other words, the outcome of one event has no effect on the outcome of another.

For example, if an individual engine fails or works, it does not impact whether another engine fails or works. This independence is significant because it allows us to calculate the probabilities of combined events more straightforwardly.

Consider these points about independent events:

• The probability of multiple independent events occurring together is the product of their individual probabilities.
• Knowing that one engine succeeds or fails does not change the probability of the other engines succeeding or failing.

In our problem, the function of each of the four airplane engines is independent. This independence allows us to use the multiplication rule for calculating the overall probability. Therefore, we are free to calculate each engine's success (functioning) probability separately, and multiply these probabilities to find the solution to our problem.

The binomial distribution is a statistical concept that describes the number of successes in a fixed number of independent trials, each with the same probability of success. It is particularly useful when the outcome of each trial can be classified as either success or failure.

The characteristics of a binomial distribution include:

• A fixed number of trials (in this case, the four engines).
• Two possible outcomes per trial (engine works or fails).
• An independent probability of each success remaining constant throughout the trials.

The binomial distribution is expressed mathematically as:
\( P(X = k) = \binom{n}{k} \, p^k \, (1-p)^{n-k} \)

Here:

• \( X \) is a random variable representing the number of successes.
• \( n \) is the number of trials (the four engines).
• \( k \) is the number of successful outcomes we are interested in.
• \( p \) is the probability of success on a single trial (0.99 engine functioning probability).

In applying the binomial distribution to our example, we focus on calculating scenarios where fewer engines fail (zero or one) compared to the entire set of four. By understanding these distributions, we efficiently identify the specific probabilities related to failing scenarios, hence assisting in determining safe possibilities.