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Probability theory is a branch of mathematics that deals with calculating the likelihood of different outcomes. It's essential for understanding how likely certain events are compared to others. In the context of our given problem, we use probability theory to assess different possibilities related to failure mechanisms in air conditioning systems.
Probability is usually expressed as a fraction or a decimal, representing how much one event might occur compared to all possible events. For example, in the solution provided, we found that the probability of a gas leak, among all systems, is computed as \( P(\text{Gas Leak}) = \frac{87}{107} \).
Conditional probability is another core aspect of probability theory and is used to find the probability of an event given that another event has already happened. Steps 3 and 4 in the solution involve conditional probabilities:

• The probability of an electrical failure, given a gas leak, was found to be \( P(\text{Electrical Failure} | \text{Gas Leak}) = \frac{55}{87} \).
• The probability of a gas leak, if we already know there's an electrical failure, was derived as \( P(\text{Gas Leak} | \text{Electrical Failure}) = \frac{55}{72} \).

Understanding probability theory helps in making informed predictions about system failures or other similar conditions.

Statistical analysis involves collecting, reviewing, and interpreting data. It's the backbone of making sense of numerous data points, like those given for the air conditioning systems.
In our scenario, we've gathered data on occurrences of gas leaks and electrical failures. The task is to use statistical methods to understand patterns and make predictions. By calculating the total number (107 systems), we gained a baseline for calculating probabilities.
Statistical analysis steps included:

• Summing up data points to determine the total sample size: 107 systems.
• Segmenting data into categories like gas leaks and electrical failures, which helps in understanding their individual probabilities.
• Performing conditional probability calculations to gain deeper insight into how events are interrelated.

By understanding such patterns, firms can better anticipate and prevent failures by focusing on the most common and critical points of failure.

Failure mechanisms refer to the specific processes that cause a system to fail. In context, it involves understanding the reasons behind air conditioning system failures. For instance, identifying whether gas leaks or electrical issues are more significant.
This exercise helps in pinpointing failure mechanisms by analyzing the data categorically:

• There are 55 systems showing both gas leaks and electrical failures.
• 32 systems have gas leaks but not electrical failures.
• 17 systems have electrical failures but no gas leaks.
• Three systems experience other issues besides the gas or electrical failures.

Understanding failure mechanisms allows maintenance firms to address the root causes of failure efficiently. This can lead to more targeted repair strategies, cost savings, and enhanced system longevity. By focusing on the most prevalent failure types, preventive measures can be implemented where they are most needed.