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Chapter 5: Problem 3

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

Expert verified
Answer: No, the number of red balls (x) is not a binomial random variable because the experiment does not meet all the characteristics of a binomial experiment. Specifically, the probability of success does not remain constant for each trial, and the trials are not independent as the balls are drawn without replacement.

Step by step solution

01

Understand the characteristics of a binomial experiment

A binomial experiment must satisfy the following conditions: 1. The experiment consists of a fixed number of trials (n). 2. Each trial has only two possible outcomes: success or failure. 3. The probability of success (p) remains constant for each trial. 4. The trials are independent, meaning the outcome of one trial does not affect the outcome of another trial.
02

Determine if the experiment meets the binomial characteristics

Let's consider each characteristic to see if our experiment meets the requirements. 1. The experiment consists of a fixed number of trials (n) - Yes, we have exactly two trials, i.e., selecting two balls from the jar. 2. Each trial has only two possible outcomes: success or failure - Yes, each trial has two outcomes: selecting a red ball (success) or selecting a white ball (failure). 3. The probability of success (p) remains constant for each trial - No, since the balls are drawn without replacement, the probability of success changes after the first trial. The probability of success in the first trial is 3/5, and if the first ball selected is red, then the probability of success in the second trial is 2/4, and if the first ball selected is white, then the probability of success in the second trial is 3/4. 4. The trials are independent, meaning the outcome of one trial does not affect the outcome of another trial - No, since the balls are drawn without replacement, the outcome of the first trial affects the probability of success in the second trial. Since the experiment does not meet all the characteristics of a binomial experiment, the number of red balls (x) is not a binomial random variable.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Experiment
A binomial experiment is a specific type of statistical experiment that must meet certain criteria. These criteria help determine whether an experiment can be classified as binomial.
Here are the key characteristics:
  • There is a fixed number of trials, denoted by \(n\). Once these trials are set, they do not change.
  • Each trial can result in only one of two possible outcomes: typically labeled as "success" or "failure."
  • The probability of success, often denoted by \(p\), remains the same across each trial.
  • The trials must be independent. This means the outcome of one trial does not affect the outcome of another.
In the context of the jar of balls, if we look at the conditions for a binomial experiment, we see that although there is a fixed number of trials (two), the trials are not independent because pulling a ball changes the number left in the jar.
Plus, the probability of pulling a red ball changes.
Thus, this experiment does not satisfy all the conditions of a binomial experiment.
Random Variable
A random variable is a numerical description of the outcomes of a statistical experiment. In our context, it's the concept used to quantify outcomes of selecting balls from a jar.
When we describe outcomes as random variables, we often denote them using \(x\), \(y\), or any alphabet letter.
Here, the random variable \(x\) represents the number of red balls selected from the jar.
Random variables can be of two types:
  • Discrete random variables, which have a countable number of outcomes.
  • Continuous random variables, which have an infinite number of possible values.
In our scenario, \(x\) is a discrete random variable because it can only take on a limited set of values: 0, 1, or 2 red balls.
This framework helps us analyze the distribution and likelihood of various outcomes.
Probability of Success
The probability of success indicates the likelihood of the desired outcome occurring in a particular trial of an experiment. In a binomial experiment, this probability is constant for each trial.
In our scenario, if we define picking a red ball as a 'success,' we observe that the probability of success is different across the draws.When the first draw occurs, the probability of drawing a red ball is \(\frac{3}{5}\).
Suppose a red ball is drawn; the probability for the second to also be a red ball falls to \(\frac{2}{4}\), since the first red ball has been removed, reducing both red balls and overall ball count.
If a white ball is first drawn, the probability changes again.
  • First red: \(\frac{2}{4}\) probability in second
  • First white: \(\frac{3}{4}\) probability in second
Because the probability of success does not remain constant, the experiment fails one of the binomial experiment criteria.
Trials without Replacement
Trials without replacement are trials in which once an item is selected from a group, it is not put back before the next selection. In our exercise, balls are selected from a jar without replacement.
This has specific implications:
  • The total number of items (balls) decreases with each selection.
  • This causes dependency between trials because removing a ball affects the composition of the remaining items.
  • It results in a changing probability of success, since the composition of the jar changes after each draw.
Conducting experiments without replacement means every draw impacts the outcomes of subsequent draws.
It contradicts the independence principle required in a binomial experiment, making this exercise unsuitable for binomial probability calculations. By fully understanding these conditions, we realize why this setup diverges from being purely binomial.

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Most popular questions from this chapter

The increased number of small commuter planes in major airports has heightened concern over air safety. An eastern airport has recorded a monthly average of five near-misses on landings and takeoffs in the past 5 years. a. Find the probability that during a given month there are no near-misses on landings and takeoffs at the airport. b. Find the probability that during a given month there are five near-misses c. Find the probability that there are at least five nearmisses during a particular month.

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