Question

Explain why \(x\) is or is not a binomial random variable. (Hint: compare the characteristics of this experiment with those of a binomial experiment given in this section.) If the experiment is binomial, give the value of \(n\) and \(p\), if possible. Two balls are randomly selected with replacement from a jar that contains three red and two white balls. The number \(x\) of red balls is recorded.

Short answer

If so, find the values of \(n\) and \(p\). Answer: Yes, the experiment is a binomial experiment. The values of \(n\) and \(p\) are \(n = 2\) trials and \(p = \frac{3}{5}\) probability of success, respectively.

Step-by-step solution

Understand the characteristics of a binomial experiment

A binomial experiment must have the following properties: 1. There must be a fixed number of trials. 2. Each trial must have only two outcomes (success or failure). 3. The trials must be independent. 4. The probability of success must remain constant throughout the trials.

Analyze the given experiment

Here, we have the experiment of randomly selecting two balls with replacement from a jar containing three red and two white balls. The number \(x\) of red balls is recorded. 1. There are a fixed number of trials (2 trials - selecting 2 balls). 2. Each trial has two outcomes (selecting a red ball or selecting a white ball). 3. Since the balls are selected with replacement, the trials are independent. 4. The probability of success (selecting a red ball) is constant because of the replacement, and thus it remains \(\frac{3}{5}\) for both trials.

Conclusion and finding values of \(n\) and \(p\)

From the analysis in Step 2, it is clear that the given experiment satisfies all the properties of a binomial experiment. Therefore, the number of red balls, \(x\), is a binomial random variable. The value of \(n\) is the number of trials; thus, \(n = 2\). The value of \(p\) is the probability of success (selecting a red ball), which is \(\frac{3}{5}\). Hence, \(p = \frac{3}{5}\).

Key concepts

Binomial Experiment Properties

When it comes to understanding binomial random variables, it is crucial to grasp the properties of a binomial experiment. By definition, a binomial experiment is one that meets certain strict criteria which ensure that the variable of interest behaves in a predictable manner. The characteristics are as follows:

• Fixed Number of Trials: The experiment must be conducted a specific and predetermined number of times. Each repetition is known as a trial.
• Two Outcomes: Each trial can only end in one of two ways, commonly referred to as 'success' and 'failure'.
• Independence of Trials: What happens in one trial does not affect the outcome of other trials. This means previous results have no bearing on future outcomes.
• Constant Probability of Success: The chance of achieving a 'success' must be the same for each trial.

In our example of selecting balls from a jar, these properties are met, which indicates a binomial experiment is present. Not only does this concept ensure consistency in the approach, but it also enables us to apply specific mathematical techniques to calculate probabilities and other statistics related to the experiment.

Probability of Success

Within the context of a binomial experiment, the 'probability of success' is a pivotal concept. To put it simply, it is the likelihood that a trial will result in the outcome designated as a success. For the formula to calculate binomial probabilities to work correctly, this probability must remain fixed from trial to trial.

In practical terms, if we consider selecting a red ball as a success in the given problem, the probability of getting a red ball is \[\begin{equation}\( p = \frac{3}{5} \)\end{equation}\]for each selection since there are 3 red balls out of a total of 5. This value is fundamental for calculating binomial probabilities, and hence understanding what 'success' means in the given context is essential. The constancy of \( p ) allows us to use binomial probability distributions, a powerful tool in statistics.

Independent Trials

Independence of trials is another cornerstone of a binomial experiment. This characteristic ensures that the outcome of one trial does not influence another. In other words, performing one trial of an experiment should have no impact on the subsequent trial's result.

The criterion of independence is fulfilled in our example by the practice of selecting balls with replacement. After a ball is chosen, it is put back into the jar thereby maintaining the original composition of the ball population for the next draw. This ensures that each draw is independent of the previous ones, a necessary condition for a variable to be binomial. It's this independence that allows for the simplification of complex probability questions into a series of more manageable calculations.