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In probability and statistics, a *random variable* represents a numerical outcome of a random process. Random variables can take on different values, each associated with a probability. In the context of our problem, the random variable \( X \) defines the count of seniors who have participated in after-school sports throughout their four years in high school, out of a total sample of 60 seniors.

Considering this, \( X \) can take any whole number value from 0 to 60. Each of these possible outcomes signifies a different number of seniors, reflecting the inherent randomness of the selection process. Therefore, recognizing the role of \( X \) as a random variable allows us to understand and describe the variability within the given problem scenario.

The *expected value* of a random variable is a key concept in statistics that represents the average or mean value it takes across numerous repeated trials or experiments. For a binomial distribution, the expected value \( E(X) \) is calculated using the formula: \( E(X) = n \times p \), where \( n \) is the number of trials, and \( p \) is the probability of success in a single trial.

In our example, we have 60 trials (seniors), and the probability that a senior participated in all four years of after-school sports is 0.08. Thus, the expected number of students who completed the activity all four years is calculated as: \[ E(X) = 60 \times 0.08 = 4.8 \].

This indicates that, on average, we can expect about 4.8 seniors in a group of 60 to have participated in after-school sports for all four years. While you cannot have a fraction of a person participate, this value provides a statistical mean for the scenario.

Calculating probabilities in a *binomial distribution* involves determining the probability of observing a particular number of successes in a sequence of independent experiments, where each experiment is a yes-no scenario. The formula used is: \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( n \) is the number of trials, \( k \) is the number of successful trials, and \( p \) is the probability of success on each trial.

For our problem, \( P(X = 0) \) tells us the probability that none of the seniors participated in after-school sports all four years, and using the formula, it is approximately 0.0125. This relatively low probability indicates that it is quite unlikely for none of the seniors to have participated.

Similarly, calculating \( P(X = 4) \) and \( P(X = 5) \) allows us to understand the likelihood of exactly four or five students participating. With probabilities of approximately 0.1863 (for \( k = 4 \)) and 0.1704 (for \( k = 5 \)), it is slightly more probable that exactly four seniors, rather than five, participated in the sports activity over the four years.

*Statistical analysis* involves the examination and interpretation of data to uncover underlying patterns and trends. In our exercise, we are using binomial distribution as a tool for analyzing the scenario of seniors' participation in after-school sports.

We determined the distribution of the random variable \( X \) as binomial: \( X \sim B(60, 0.08) \). This choice captures the core features of the problem: a fixed number of trials, binary outcomes (participate or not), constant probability of success, and independent trials.

Using the distribution, we perform calculations to answer questions like the expected number of participating seniors and probabilities of zero, four, or five seniors participating. Such analysis is crucial in understanding not just the immediate results but also the broader statistical performance and reliability of the data. Applying statistical tools in this structured manner enhances our ability to make informed predictions and judgments based on numerical evidence.