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Chapter 5: Problem 59

List the five identifying characteristics of the binomial experiment.

Short Answer

Expert verified
Answer: The five identifying characteristics of a binomial experiment are: 1) Two possible outcomes (success and failure), 2) fixed number of trials, 3) independent trials, 4) constant probability of success, and 5) randomly conducted trials.

Step by step solution

01

1. Two possible outcomes

In a binomial experiment, each trial has only two possible outcomes, typically labeled as success (S) and failure (F). The outcome probability remains constant throughout the experiment.
02

2. Fixed number of trials

The experiment consists of a fixed number of trials (n), and each trial is independent of the others. That is, the outcome of one trial does not affect the outcome of any other trial.
03

3. Independent trials

The outcomes of each trial are statistically independent, meaning that the outcome of one trial does not influence the outcome of another trial. In a binomial experiment, the probability of success (p) and the probability of failure (q) remain constant for all trials.
04

4. Constant probability of success

In a binomial experiment, the probability of success (p) remains constant for all trials. The probability of success on any single trial is the same each time the experiment is run.
05

5. Randomly conducted trials

The trials in a binomial experiment are conducted randomly. This means that there is no bias or pattern influencing the outcomes of the trials, ensuring the validity of the statistical analysis.

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Key Concepts

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

Two Possible Outcomes
In a binomial experiment, each trial presents exactly two distinct outcomes. Think of them as success and failure. For example, when you flip a coin, there's a heads and a tails outcome. This binary nature of outcomes makes binomial experiments straightforward to analyze. Both outcomes, success and failure, are mutually exclusive, meaning only one can occur in any given trial. This distinct separation helps in simplifying calculations and predictions about the experiment. While the specific labels (success or failure) can vary depending on the context, what truly defines a binomial experiment is that only two outcomes exist for every single trial conducted.
Fixed Number of Trials
A key feature of a binomial experiment is that it involves a fixed number of trials. This means you decide beforehand how many times to repeat the experiment or process. For instance, if you decide to flip a coin 10 times, that number "10" becomes your fixed number of trials. Each of these trials is pre-determined and non-negotiable once the experiment begins. Knowing the number of trials in advance allows for precise calculation of probabilities, as it sets a clear framework for the experiment. With fixed trials, you can predict outcomes and assess likelihoods with greater confidence, using the binomial probability formula.
Independent Trials
One hallmark of a binomial experiment is the independence of each trial. This means the outcome of one trial has no effect whatsoever on the outcome of another. Consider flipping a coin: whether the result is heads or tails on one flip does not change the outcome of the next flip. This independence is crucial because it ensures that the probabilities remain consistent throughout the experiment. Each trial stands alone, contributing to the overall results without interfering with other trials. Maintaining independence in trials is essential for ensuring that calculations and predictions remain accurate and unbiased.
Constant Probability of Success
In a binomial experiment, the probability of success ( p ) is constant throughout all trials. This means if the likelihood of rolling a die and getting a "6" is 1/6 on the first roll, it remains 1/6 on every subsequent roll. Consistency in the probability of success is crucial because it ensures that statistical analysis remains straightforward, without having to adjust for any changes in probability. With a constant probability, you can utilize the binomial probability formula to calculate outcomes and make predictions confidently. This stability is what allows binomial models to be effectively used in various fields such as genetics, quality control, and finance.

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Most popular questions from this chapter

Under what conditions can the Poisson random variable be used to approximate the probabilities associated with the binomial random variable? What application does the Poisson distribution have other than to estimate certain binomial probabilities?

The 10 -year survival rate for bladder cancer is approximately \(50 \%\). If 20 people who have bladder cancer are properly treated for the disease, what is the probability that: a. At least 1 will survive for 10 years? b. At least 10 will survive for 10 years? c. At least 15 will survive for 10 years?

The number \(x\) of people entering the intensive care unit at a particular hospital on any one day has a Poisson probability distribution with mean equal to five persons per day. a. What is the probability that the number of people entering the intensive care unit on a particular day is two? Less than or equal to two? b. Is it likely that \(x\) will exceed \(10 ?\) Explain.

Refer to Exercise 5.27 where \(30 \%\) of all admitted patients fail to pay their bills and the debts are eventually forgiven. Suppose that the clinic treats 2000 different patients over a period of 1 year, and let \(x\) be the number of forgiven debts. a. What is the mean (expected) number of debts that have to be forgiven? b. Find the variance and standard deviation of \(x\). c. What can you say about the probability that \(x\) will exceed \(700 ?\) (HINT: Use the values of \(\mu\) and \(\sigma\), along with Tchebysheff's Theorem.)

If \(x\) has a binomial distribution with \(p=.5\), will the shape of the probability distribution be symmetric, skewed to the left, or skewed to the right?
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