Question

Class Year Suppose that undergraduate students at a university are equally divided between the four class years (first-year, sophomore, junior, senior) so that the probability of a randomly chosen student being in any one of the years is \(0.25 .\) If we randomly select four students, give the probability function for each value of the random variable \(X=\) the number of seniors in the four students.

Short answer

The probability function for each value of the random variable \(X =\) the number of seniors in the four students would be found by using the binomial probability formula for each value of \(X\). The results of the calculations would be the probability for each value of \(X\)

Step-by-step solution

Understanding the situation and variables

We are trying to choose a number of specific class years, seniors, from a larger group. Let the random variable \(X\) be the number of seniors chosen. Possible values of \(X\) are \(0, 1, 2, 3\), and \(4\), because we are choosing four students.

Applying the binomial probability formula for each value of X

In this case, the Binomial Probability Formula is used, which is: \[P(x; n, p) = \binom{n}{x} * (p^{x}) * ((1 - p)^{n - x})\], where \(p\) is the probability of picking a senior student (0.25), \(n\) is the number of trials (4 students), and \(x\) is the specific number of seniors we're testing the probability for.

Calculate probability for \(X = 0\)

Substitute \(0\) for \(x\) into the formula, this will give you the probability of picking no senior students: \[P(0; 4, 0.25) = \binom{4}{0} * (0.25^{0}) * ((1 - 0.25)^{4 - 0})\]

Calculate probability for \(X = 1\)

Substitute \(1\) for \(x\) into the formula, this will give you the probability of picking one senior student: \[P(1; 4, 0.25) = \binom{4}{1} * (0.25^{1}) * ((1 - 0.25)^{4 - 1})\]

Calculate probability for \(X = 2\)

Substitute \(2\) for \(x\) into the formula, this will give you the probability of picking two senior students: \[P(2; 4, 0.25) = \binom{4}{2} * (0.25^{2}) * ((1 - 0.25)^{4 - 2})\]

Calculate probability for \(X = 3\)

Substitute \(3\) for \(x\) into the formula, this will give you the probability of picking three senior students: \[P(3; 4, 0.25) = \binom{4}{3} * (0.25^{3}) * ((1 - 0.25)^{4 - 3})\]

Calculate probability for \(X = 4\)

Substitute \(4\) for \(x\) into the formula, this will give you the probability of picking four senior students: \[P(4; 4, 0.25) = \binom{4}{4} * (0.25^{4}) * ((1 - 0.25)^{0})\]

Key concepts

Understanding Probability Functions

A probability function is a mathematical expression that gives the probabilities of occurrence of different possible outcomes for a random experiment. It assigns a probability to each outcome in the space of possible outcomes, thereby providing a complete description of the random process in question.

In the exercise provided, the students are asked to find the probability function for the random variable representing the count of seniors in a selection of four students. Since there are equal chances for a student to belong to any of the class years, the probability of selecting a senior for each trial is 0.25. The probability function for this scenario would therefore assign the likelihood that 0, 1, 2, 3, or 4 seniors are selected out of the four students.

By definition, a probability function must satisfy two conditions: the sum of probabilities of all possible outcomes must be 1, and each individual probability must be between 0 and 1, inclusive. This ensures that probabilities are properly normalized and reflect the true nature of a random process.

Random Variables and Their Role

A random variable is a numerical description of the outcome of a statistical experiment. It is a variable that takes on different values randomly. In our example, the random variable is denoted by \(X\), representing the number of seniors in a randomly selected group of four students.

The values that \(X\) can take - 0, 1, 2, 3, or 4 - are determined by the number of possible seniors that can appear in the selection. This concept is critical for understanding the formulation of probability problems and allows for the systematic calculation of the probabilities of complex events.

The use of a random variable simplifies the process of finding probabilities because it encapsulates all possible outcomes into a single, quantifiable metric. It also paves the way for the application of probability distributions, like the binomial distribution, which provide a framework for calculating the likelihood of different outcomes.

Demystifying the Binomial Distribution

The binomial distribution is one of the most fundamental probability distributions in statistics. It models the number of successes in a fixed number of independent trials, with each trial having the same probability of success.

In the context of our exercise, the binomial distribution is used to determine the probability of selecting a certain number of seniors from the four students. Since each selection is independent and the probability for selecting a senior (success) remains constant at 0.25, this scenario perfectly fits the criteria for using the binomial distribution formula provided in the steps.

The formula incorporates factorials and powers to calculate the likelihood of each possible count of seniors. It is expressed as:\[P(x; n, p) = \binom{n}{x} * (p^{x}) * ((1 - p)^{n - x})\], where \(n\) is the total number of trials, \(p\) the probability of success on any given trial, and \(x\) the number of successes we want to find the probability for.

Using this formula, predictions about the composition of randomly created groups or outcomes of repeated, independent events can be made with confidence and accuracy. The binomial distribution is a powerful tool in probability and statistics, and understanding it is essential for interpreting results in various fields.