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The probability distribution function (PDF) is a fundamental concept in statistics, especially when dealing with random variables. It provides a way to describe how probabilities are distributed over the values that a random variable can take. In our case, the random variable is the number of students (out of eight) who believe same-sex couples should have the right to marry.

The PDF is derived under the consideration of a certain type of distribution—in this case, a binomial distribution. This is because we perform eight independent trials (each trial representing a student's opinion), which have only two possible outcomes: either the student agrees or disagrees. By calculating the probability of each possible outcome from zero to eight agreeing students, we can tabulate these probabilities to form the PDF.

• The PDF table will show each possible outcome for the number of students.
• For instance, it contains rows for zero students agreeing, one student agreeing, and so on, up to all eight students agreeing.
• The value next to each outcome denotes the probability of that outcome occurring, as calculated from the binomial probability formula.

The binomial probability formula is the key to calculating the probability of a given number of successes in a series of independent trials. It's perfect for situations like the one we're examining, where each trial has only two possible outcomes. The formula is given by:\[ P(x) = \binom{n}{x} p^x (1-p)^{n-x} \]Here, we need to understand what each part of the formula signifies:

• \( n \) is the total number of trials, which is the number of students in this example (8).
• \( x \) is the number of successful trials, which means the number of students agreeing.
• \( p \) is the probability of success in a single trial (0.713 for a student agreeing).
• \( (1-p) \) is the probability of failure, i.e., a student disagreeing (0.287).

Each probability value computed using this expression provides the chance that exactly \( x \) students out of \( n \) will agree. By evaluating this expression for \( x = 0 \) through \( x = n \), one can fill in all the values of the probability distribution function table.

A crucial component of the binomial probability formula is the binomial coefficient, denoted as \( \binom{n}{x} \). It represents the number of ways to choose \( x \) successes from \( n \) trials, acting as a multiplier for probability. Understanding this can simplify solving problems involving binomial distributions. The binomial coefficient is calculated using the formula:\[ \binom{n}{x} = \frac{n!}{x!(n-x)!} \]In this formula:

• \(!\) indicates a factorial, meaning the product of all positive integers up to that number. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
• \( n! \) is the factorial of the total number of trials.
• \( x! \) is the factorial of the number of successful outcomes.
• \((n-x)!\) is the factorial of the number of unsuccessful trials.

The binomial coefficient essentially helps in determining all possible ways \( x \) successes can appear in \( n \) trials, which is essential when calculating the probability for each outcome in the PDF.