Question

A jar contains five balls: three red and two white. Two balls are randomly selected without replacement from the jar, and the number \(x\) of red balls is recorded. Explain why \(x\) is or is not a binomial random variable. (HINT: Compare the characteristics of this experiment with the characteristics of a binomial experiment given in this section.) If the experiment is binomial, give the values of \(n\) and \(p\).

Short answer

Answer: No, the number of red balls selected (x) in this experiment is not a binomial random variable because the trials are not independent and the probability of success is not the same in every trial.

Step-by-step solution

Understanding the problem

Before we delve into analyzing the exercise, let's first understand what a binomial experiment is. A binomial experiment is a statistical experiment with the following characteristics: 1. The experiment consists of \(n\) independent trials. 2. There are only two possible outcomes in each trial, often called "success" and "failure." 3. The probability of success (denoted as \(p\)) is the same in every trial. Now, let's analyze our jar problem and compare it with the characteristics of a binomial experiment.

Comparing with binomial experiment characteristics

In this exercise, we are selecting two balls from the jar, so we have \(n = 2\) trials. Now, let's compare the experiment with the characteristics of a binomial experiment: 1. Are the trials independent? Since the balls are selected without replacement, the outcome of the first trial affects the probability in the second trial. So, the trials are not independent. 2. Are there only two possible outcomes, i.e., success and failure, in each trial? Yes, there are two possible outcomes: selecting a red ball or not. 3. Is the probability of success the same in every trial? In the first trial, the probability of drawing a red ball is \(\frac{3}{5}\), while in the second trial, it changes to either \(\frac{2}{4}\) or \(\frac{3}{4}\) based on the result of the first trial. Therefore, the probability of success is not the same in every trial.

Conclusion

Since the trials are not independent and the probability of success is not the same in every trial, we can conclude that the number of red balls selected (\(x\)) in this experiment is not a binomial random variable. Therefore, we won't provide the values of \(n\) and \(p\) as the experiment is not binomial.

Key concepts

Binomial Experiment

A binomial experiment is a specific type of experiment in probability theory. It is characterized by several key features that make it unique and identifiable. To see if an experiment fits the binomial model, it should follow these three main criteria:

• The experiment involves a fixed number of trials, denoted as \( n \).
• Each trial is independent of the others.
• Every trial results in one of two outcomes: success or failure.
• The probability of success, often represented as \( p \), remains constant throughout all trials.

Breaking these down, you can spot the distinction made by these criteria in any given scenario. For example, if a situation involves selecting items from a pool without changing the probability of subsequent selections, it deviates from the standard binomial setup.

Independent Trials

In a binomial experiment, it's crucial that each trial is independent. Meaning, the result of one trial should not influence the result of another. This is somewhat like flipping a coin—each flip doesn't affect the next.
When trials are independent:

• The probability of success remains the same because one trial doesn’t alter the conditions of the next.
• The outcomes are isolated, ensuring the purity of the experimental design.

In the exercise with balls being drawn without replacement, the trials affect one another. The outcome from drawing the first ball influences the likelihood of outcomes in the second draw, making them dependent. This lack of independence disqualifies the experiment from being truly binomial.

Probability of Success

The probability of success, denoted \( p \), is integral to defining a binomial experiment. The constancy of this probability across trials is what sets binomial experiments apart.
Consider an experiment where the probability of success shifts with each trial; this is not binomial.
Let's relate this idea back to our exercise:

• The probability of selecting a red ball in the first draw is \( \frac{3}{5} \).
• In the second draw, probability changes to \( \frac{2}{4} \) or \( \frac{3}{4} \) depending on the first draw's result.

This variation violates the binomial experiment principle where \( p \) should be constant. In summary, when the probability of success isn’t stable across trials, the experiment does not align with binomial conditions.