Question

In Exercises 34-37, 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 without replacement from a jar that contains three red and two white balls. The number \(x\) of red balls is recorded.

Short answer

Answer: No, x is not a binomial random variable in this experiment, because the probability of success is not constant, and the trials are dependent.

Step-by-step solution

1. Fixed number of trials

The experiment consists of selecting two balls from the jar, so there are a fixed number of trials (n = 2).

2. Success or failure classification

The outcome of each trial can be classified as a 'success' if we draw a red ball and a 'failure' if we draw a white ball.

3. Constant probability of success

Since we are selecting the balls without replacement, the probability of success (drawing a red ball) changes after each draw. Initially, the probability p1 of success (drawing a red ball) is 3/5 (three red balls out of a total of five). After drawing the first ball, the probability p2 of success (again drawing a red ball) is either 2/4 (if the first ball was red) or 3/4 (if the first ball was white). Since the probability of success changes depending on the first draw, the probability is not constant.

4. Independent trials

As the balls are selected without replacement, the outcome of the first trial affects the outcome of the second trial. Therefore, the trials are dependent, not independent. Since the probability of success is not constant, and the trials are dependent, the given experiment is not a binomial experiment. So, \(x\) is not a binomial random variable.

Key concepts

Probability

Understanding the wide array of concepts in probability is crucial for analyzing experiments, like the one involving the selection of balls from a jar. Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. When we discuss the probability of drawing a red ball from a jar, we are referring to how likely that event is to happen given the setup of the experiment.

For example, if a jar contains three red balls and two white, the initial probability of drawing a red ball (\(p_1\)) is calculated by dividing the number of red balls by the total number of balls, which is \(\frac{3}{5}\). It is this numerical representation of likelihood that is the cornerstone of a multitude of probability problems and serves as a bedrock in the fields of statistics and gambling, insurance, finance, science, and many others.

Furthermore, we must consider the context of the experiment to understand how the probability may change over time, particularly in scenarios where items are selected without replacement, affecting the odds of successive events.

Binomial Experiment

A binomial experiment is a statistical experiment that satisfies several strict conditions: it must consist of a fixed number of trials, each trial must have only two possible outcomes (often classified as 'success' or 'failure'), the probability of success must remain constant across trials, and all trials must be independent of each other.

In our example with the jar of balls, while the experiment begins with a fixed number of trials, two crucial conditions are violated. Firstly, the probability of drawing a red ball is not constant because it changes after each draw. Initially, it is \(\frac{3}{5}\), but it may drop to \(\frac{2}{4}\) or rise to \(\frac{3}{4}\) after the first draw, depending on whether a red or a white ball was selected. Due to this change in probability, the experiment cannot be classified as binomial.

Secondly, because the selection is performed without replacement, the outcomes of the trials are not independent; the first draw directly influences the second. A true binomial experiment would require that each draw is not affected by previous draws, such as drawing with replacement or having a separate jar of balls for each trial.

Independent Trials

Independent trials are a pillar of particular probability exercises, specifically binomial experiments. The notion of independence between trials means that the outcome of one trial has no effect on the outcome of another. For example, flipping a coin represents independent trials because the result of one flip does not influence the results of subsequent flips.

In situations where you draw items from a set without replacement, such as the balls in our exercise, the trials are not independent. The first draw changes the composition of the jar, directly affecting the probability of the outcome in the second trial. If a red ball is drawn first, there are fewer red balls to draw from in the second trial. Contrast this with flipping a coin, where no matter how many times you flip, the probability of heads or tails remains 50/50.

Understanding whether trials are independent is essential; it determines which statistical methods and probability models are appropriate for analyzing the situation. In scenarios where trials are not independent, other probability models, like hypergeometric or conditional probability, might be more suitable to use.