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Probability theory helps us understand the likelihood of various outcomes in a random experiment. In this exercise, we're dealing with the selection of workers and calculating the probabilities of different scenarios. The key measure in probability is the likelihood or chance of an event occurring, expressed between 0 and 1. A probability of 0 means an event cannot happen, while 1 means it is certain to happen.
When looking at the problem of selecting 5 workers from a group of 24, we are interested in scenarios like all selected workers being from a particular shift. By using probability theory, we can determine the chance of such events using the formula:
\[ P( ext{Event}) = \frac{ ext{Number of favorable outcomes}}{ ext{Total number of possible outcomes}} \]Understanding probability theory requires being familiar with concepts such as:

• Sample Space: The set of all possible outcomes. Here, it is any combination of 5 workers chosen from 24.
• Events: These are one or more outcomes we are interested in — like "all workers from the day shift".

The binomial coefficient, often noted as \( \binom{n}{k} \), represents the number of ways to choose \( k \) items from \( n \) items without regard to order. This concept is crucial in combinatorics, as it provides a way to count the possible selections of workers.
In our problem, we use the binomial coefficient to find how many combinations exist in different selection scenarios, such as picking 5 out of 10 day shift workers:
\[ \binom{10}{5} = \frac{10!}{5!(10-5)!} \]The factorial function \( n! \) means the product of all positive integers up to \( n \), giving us a way to calculate these large numbers easily. It helps students understand how to distribute items in groups without caring about the order.
The binomial coefficients are essential when dealing with sampling problems, as they allow us to calculate the number of favorable ways to select certain groups effectively.

Sampling without replacement involves picking items from a pool where each item is not returned to the pool once picked. This type of sampling affects the probabilities because the pool of potential selections becomes smaller with each draw.
In the context of this exercise, when 5 workers are selected for interviews, each one selected reduces the pool of remaining workers. This means the choice of each subsequent worker affects the probability of future selections, in contrast to sampling with replacement where the pool size remains unchanged.
Key points of sampling without replacement include:

• Dependent Events: The probabilities change with each selection because the total number of workers decreases.
• No Repetition: Once a worker is selected, they cannot be chosen again for that particular sampling.

Understanding this concept is crucial for calculating probabilities in situations where the order and composition of selections matter.