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Chapter 5: Problem 4

Statistical Literacy In a binomial experiment, is it possible for the probability of success to change from one trial to the next? Explain.

Short Answer

Expert verified
In a binomial experiment, the probability of success is constant and does not change from one trial to the next due to the experiment's independent trial design.

Step by step solution

01

Understand the Binomial Experiment

A binomial experiment consists of a fixed number of trials, each with two possible outcomes: success or failure. It is important to note that each trial is independent of the others, meaning the outcome of one trial does not affect the outcome of another.
02

Determine the Probability Characteristics

In a binomial experiment, one of the key characteristics is that the probability of success remains constant across all trials. This does not change from one trial to the next.
03

Explanation of Probability Stability

Since the probability of success remains unchanged in each trial, the variable representing whether or not an individual trial is a success is only influenced by the designed conditions of the experiment and not by the outcomes of previous trials.

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Key Concepts

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

Probability in a Binomial Experiment
In a binomial experiment, the probability of success is a key element. It refers to the likelihood that a single trial within the experiment results in a favorable outcome. A binomial trial involves two possibilities: success or failure. For instance, flipping a fair coin yields heads (success) or tails (failure) with each toss possessing a fixed probability of 0.5 for heads. This value indicates the probability of success.
What's critical in a binomial experiment is that this probability remains unchanged across all trials. This consistency ensures that the statistical model remains valid, allowing us to apply the binomial probability formula accurately. In simple terms, if you know the probability of success for any one trial, you know it for all trials in the experiment due to their identical nature.
Understanding Independent Trials
Independent trials are central to the functioning of a binomial experiment. **Independence** here refers to how the outcome of one trial does not influence or alter the probabilities of subsequent trials. If one trial in an experiment results in success, this does not increase or decrease the likelihood of success in the following trial.
The concept of independent trials is crucial because:
  • Each trial is a reset of circumstances, maintaining the uniformity of conditions.
  • Errors or external influences affecting one trial don't spill over to others.
  • This independence safeguards the integrity of probabilistic calculations, ensuring accuracy.
Without this independence, the trials could be considered conditional, fundamentally altering the experiment's nature and affecting predictive outcomes.
Enhancing Statistical Literacy
Understanding the principles behind binomial experiments boosts statistical literacy immensely. Recognizing the importance of consistent probabilities and independent trials aids in grasping broader statistical concepts and methodologies.
Statistical literacy empowers one to:
  • Interpret data accurately and make informed decisions based on statistics.
  • Apply statistical methods correctly across various scenarios.
  • Communicate statistical findings clearly, enhancing analytical discussions.
By understanding these foundational concepts, individuals can better appreciate how statistical models are built and why certain protocols are essential. Such literacy forms the backbone of effective decision-making driven by data.

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Most popular questions from this chapter

Health Care: Office Visits What is the age distribution of patients who make office visits to a doctor or nurse? The following table is based on information taken from the Medical Practice Characteristics section of the Statistical Abstract of the United States (116th Edition). \begin{tabular}{l|ccccc} \hline Age group, years & Under 15 & \(15-24\) & \(25-44\) & \(45-64\) & 65 and older \\ \hline Percent of office visitors & \(20 \%\) & \(10 \%\) & \(25 \%\) & \(20 \%\) & \(25 \%\) \\ \hline \end{tabular} Suppose you are a district manager of a health management organization (HMO) that is monitoring the office of a local doctor or nurse in general family practice. This morning the office you are monitoring has eight office visits on the schedule. What is the probability that (a) at least half the patients are under 15 years old? First, explain how this can be modeled as a binomial distribution with 8 trials, where success is visitor age is under 15 years old and the probability of success is \(20 \%\). (b) from 2 to 5 patients are 65 years old or older (include 2 and 5 )? (c) from 2 to 5 patients are 45 years old or older (include 2 and 5 )? Hint: Success is 45 or older. Use the table to compute the probability of success on a single trial. (d) all the patients are under 25 years of age? (e) all the patients are 15 years old or older?

Quota Problem: Archaeology An archaeological excavation at Burnt Mesa Pueblo showed that about \(10 \%\) of the flaked stone objects were finished arrow points (Source: Bandelier Archaeological Excavation Project: Summer 1990 Excavations at Burnt Mesa Pueblo, edited by Kohler, Washington State University). How many flaked stone objects need to be found to be \(90 \%\) sure that at least one is a finished arrow point? Hint: Use a calculator and note that \(P(r \geq 1) \geq 0.90\) is equivalent to \(1-P(0) \geq 0.90\), or \(P(0) \geq 0.10 .\)

Binomial Probabilities: Coin Flip A fair quarter is flipped three times. For each of the following probabilities, use the formula for the binomial distribution and a calculator to compute the requested probability. Next, look up the probability in Table 3 of Appendix II and compare the table result with the computed result. (a) Find the probability of getting exactly three heads. (b) Find the probability of getting exactly two heads. (c) Find the probability of getting two or more heads. (d) Find the probability of getting exactly three tails.

Agriculture: Apples Approximately \(3.6 \%\) of all (untreated) Jonathan apples had bitter pit in a study conducted by the botanists Ratkowsky and Martin (Source: Australian Journal of Agricultural Research, Vol. 25, pp. \(783-790)\). (Bitter pit is a disease of apples resulting in a soggy core, which can be caused either by overwatering the apple tree or by a calcium deficiency in the soil.) Let \(n\) be a random variable that represents the first Jonathan apple chosen at random that has bitter pit. (a) Write out a formula for the probability distribution of the random variable \(n\). (b) Find the probabilities that \(n=3, n=5\), and \(n=12\). (c) Find the probability that \(n \geq 5\). (d) What is the expected number of apples that must be examined to find the first one with bitter pit? Hint: Use \(\mu\) for the geometric distribution and round.

Basic Computation: Geometric Distribution Given a binomial experiment with probability of success on a single trial \(p=0.30\), find the probability that the first success occurs on trial number \(n=2\).
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