91Ó°ÊÓ

A random variable is a fundamental concept in probability and statistics. It represents a variable that can take on different possible values, each with an associated probability. In the context of the exercise, the random variable \(X\) is defined as the number of Shotokan Karate students selected for the demonstration.

To illustrate this, consider the pool of students available: 6 are Tae Kwon Do students, and 7 are Shotokan Karate students. We are interested in how many of these Shotokan students appear in the randomly chosen group of eight.

This variation is what makes \(X\) a random variable. Its values are dependent on the outcome of the selection process where each different configuration of students means \(X\) can be a different number.

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes of a random variable.

In the case of the exercise, the distribution governing \(X\), the number of Shotokan Karate students, is a hypergeometric distribution. We use a hypergeometric distribution because our selection is made without replacement from a finite population. This means once a student is selected, they are not returned to the pool for subsequent selections, which accurately reflects the setup of the exercise.

The hypergeometric probability distribution is defined by three parameters:

• \(N = 13\): the total number of students.
• \(K = 7\): the total number of Shotokan Karate students.
• \(n = 8\): the number of students selected for the demonstration.

For each possible number of Shotokan Karate students in the group, the hypergeometric probability distribution tells us the likelihood of that number appearing.

The expected value of a random variable gives a measure of the center of the distribution of the variable. It indicates the average result we would anticipate if we could repeat the selection process over multiple trials.

For a hypergeometric distribution, the expected value \(E(X)\) of the random variable \(X\) is calculated using the formula:

\[E(X) = n \left( \frac{K}{N} \right)\]

This means for our selection of students, substituting the known values into the formula:

• \(n = 8\)
• \(K = 7\)
• \(N = 13\)

We compute:\[E(X) = 8 \left( \frac{7}{13} \right) \approx 4.31\]

Thus, we expect approximately 4 Shotokan Karate students to be included in the demonstration.

The phrase "selection without replacement" describes a process where once an item is chosen from a population, it is not replaced back into the pool for further selections. This is a critical factor for the exercise because it influences how probabilities are calculated.

In the demonstration example, each student selected reduces the available pool, and consequently changes the probabilities for subsequent selections. For instance, as you choose one Shotokan Karate student, the likelihood of picking another is reduced because the total number of Shotokan students (\(K\)) as well as the entire pool (\(N\)) decrease.

This contrasts with "selection with replacement", where the item is returned back to the population after each selection, keeping probabilities constant across picks. Understanding the implications of selecting without replacement is key to applying the hypergeometric distribution correctly in real-world scenarios like this one.