The lifespans, \(L\), of 2500 components were monitored and recorded in Table 2 . $$ \begin{array}{rr} \hline \text { Lifespan, } L \text { (hours) } & \text { Frequency } \\ \hline 0 \leq L \leq 5000 & 16 \\ 500012000 & 145 \\ \hline \end{array} $$ Calculate the probability that a component picked at random has a lifespan (a) of between 8000 and 11000 hours (b) of more than 11000 hours (c) of more than 10000 hours given that it has already lasted for at least 8000 hours (d) of between 11000 and 12000 hours given that it has already lasted for at least 9000 hours.

Lifespan distribution in a dataset highlights the different lengths of time for which components, such as electronic parts or mechanical elements, remain operational before failing. It is a vital aspect of reliability engineering and helps in predicting the longevity and performance of components over a given period. In our case, the lifespan distribution is represented through a frequency distribution table that displays the number of components that last within specific lifespan intervals.

Understanding lifespan distribution is particularly important for manufacturing and warranty analysis, as it can help estimate the percentage of components expected to fail within a certain time frame. The distribution can follow various statistical models, such as exponential, normal, or Weibull, depending on the nature of the components and their failure mechanisms.

Conditional probability is a crucial concept in probability theory that deals with the likelihood of an event occurring given that another event has already happened. It is expressed as P(A|B), where A and B are two events and P(A|B) is the probability of event A occurring given that event B is known to occur. For example, in the given problem, if we're looking at the probability of a component lasting more than 10000 hours given it has already lasted at least 8000 hours, we're dealing with conditional probability.

This concept is extensively used in a variety of fields such as finance, healthcare, and engineering for risk assessment and decision-making processes. It helps in updating the probabilities of outcomes when new information becomes available. For students, a solid grasp of conditional probability is essential for understanding more complex statistical methods and analyses.

A frequency distribution table is a simple way to present statistical data. It organizes a set of data points by specifying the frequency, or how often each value occurs in the dataset. In our problem, the 'lifespan' is the set of data points with the frequency indicating the number of components having lifespans within specific intervals.

Frequency distribution tables are essential for visualizing data patterns, identifying the mode, and simplifying the calculation of probabilities. The table in the given exercise allows us to quickly assess the range of lifespans and their corresponding frequencies. This direct form of data representation helps in making probability calculations for certain events much more straightforward.

The concept of a random component lifespan refers to the notion that the life expectancy of any individual unit within a batch of components cannot be determined with certainty. Each component has a chance or 'probability' of lasting for a certain amount of time before failure. This randomness is inherently tied to factors such as the quality of materials, environmental conditions, manufacturing processes, and usage patterns.

In predictive maintenance and reliability engineering, understanding this randomness is key to estimating maintenance schedules and the need for replacements. Using the provided frequency distribution, we created a probabilistic model that can give us insights into the overall reliability of a batch of these components. Insight into the random lifespan of these components allows businesses to better manage inventory and forecast potential issues.