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Chapter 5: Problem 2

What does it mean to say that the trials of an experiment are independent?

Short Answer

Expert verified
Independent trials mean the outcome of one trial doesn't affect another's.

Step by step solution

01

Understand the Concept of a Trial

In probability and statistics, a trial refers to a single occurrence or performance of an experiment. For instance, rolling a die once or flipping a coin is considered a trial. When experiments are repeated, each repetition is known as a trial.
02

Define Independence in Probability

Two events are independent if the occurrence of one event does not affect the probability of the occurrence of the other event. Mathematically, events A and B are independent if \( P(A \cap B) = P(A) \times P(B) \), where \( P(A \cap B) \) is the probability that both events happen.
03

Apply Independence to Trials

In the context of trials, saying that the trials are independent means that the outcome of one trial does not influence the outcome of another trial. For example, flipping a coin does not affect the next coin flip's outcome; each flip is independent of the previous ones.
04

Examples of Independent Trials

Consider rolling a die multiple times: each roll is independent of the others because the die does not "remember" previous results. Similarly, drawing replacement cards from a deck involves independent trials, as each draw is unaffected by previous ones after replacement.

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Key Concepts

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

Probability
Probability is the foundation upon which much of statistics is built. In simple terms, it measures how likely an event is to occur, expressed as a number between 0 and 1.
A probability of 0 means the event won't happen, while a probability of 1 means it definitely will. Probability helps us predict the likelihood of an occurrence even when there is an element of randomness involved.
  • For example, when you flip a fair coin, there are two possible outcomes: heads or tails. Each outcome has a probability of 0.5, assuming the coin is balanced and flipped fairly.
  • Probability can be expressed as a fraction, a decimal, or a percentage, making it versatile in its applicability.
  • Understanding the concept of probability allows us to make informed decisions based on the likelihood of various outcomes.
It serves as a crucial tool in a variety of fields, from predicting weather patterns to calculating insurance risks.
Statistical Independence
Statistical Independence is a vital concept in the study of probability and statistics. It describes the scenario when two or more events do not affect each other鈥檚 occurrence.
  • Whenever two events are independent, the probability of them both occurring is the product of their individual probabilities.
  • This is mathematically represented as: if events A and B are independent, then \( P(A \cap B) = P(A) \times P(B) \).
  • A common real-world example includes the flip of a coin and the roll of a die happening consecutively. The result of one does not impact the other.
Understanding whether events are independent helps us simplify complex probability problems and provides clarity in the analysis of data sets.
Experiments and Sampling
In probability and statistics, experiments refer to any process that leads to a result or outcome, like rolling a die or conducting a survey. Each performance or replication of such an experiment is known as a trial.
Sampling involves selecting a part or segment of the population to conduct an experiment on, and there are several ways to sample.
  • "With replacement" means that each item is returned to the population before the next trial, ensuring independent trials.
  • "Without replacement" can lead to dependent trials because each selection affects the remaining population.
  • Sampling methods affect the independence of trials and therefore the interpretation of results.
Understanding how to design experiments and choose appropriate sampling techniques is crucial for drawing valid conclusions in studies.
Probability of Events
The probability of events focuses on figuring out how likely various specific outcomes are within a sample space 鈥� the set of all possible outcomes.
  • An event might be singular, like rolling a 3 on a die, or compound, like rolling an even number.
  • For independent events, the probability the compound event happens is the product of the probabilities of each independent event.
  • For example, the probability of rolling a 3 and then an even number with a die is \( \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} \).
By calculating these probabilities, individuals can assess risk, make predictions, and even strategize effectively in games or decision-making processes.

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Most popular questions from this chapter

In the western United States, there are many dry-land wheat farms that depend on winter snow and spring rain to produce good crops. About \(65 \%\) of the years, there is enough moisture to produce a good wheat crop, depending on the region (Reference: Agricultural Statistics, U.S. Department of Agriculture). (a) Let \(r\) be a random variable that represents the number of good wheat crops in \(n=8\) years. Suppose the Zimmer farm has reason to believe that at least 4 out of 8 years will be good. However, they need at least 6 good years out of 8 to survive financially. Compute the probability that the Zimmers will get at least 6 good years out of \(8,\) given what they believe is true; that is, compute \(P(6 \leq r | 4 \leq r) .\) See part (d) for a hint.

Henry Petroski is a professor of civil engineering at Duke University. In his book To Engineer Is Human: The Role of Failure in Successful Design, Professor Petroski says that up to \(95 \%\) of all structural failures, including those of bridges, airplanes, and other commonplace products of technology, are believed to be the result of crack growth. In most cases, the cracks grow slowly. It is only when the cracks reach intolerable proportions and still go undetected that catastrophe can occur. In a cement retaining wall, occasional hairline cracks are normal and nothing to worry about. If these cracks are spread out and not too close together, the wall is considered safe. However, if a number of cracks group together in a small region, there may be real trouble. Suppose a given cement retaining wall is considered safe if hairline cracks are evenly spread out and occur on the average of 4.2 cracks per 30 -foot section of wall. (a) Explain why a Poisson probability distribution would be a good choice for the random variable \(r=\) number of hairline cracks for a given length of retaining wall. (b) In a 50 -foot section of safe wall, what is the probability of three (evenly spread-out) hairline cracks? What is the probability of three or more (evenly spread-out) hairline cracks? (c) Answer part (b) for a 20 -foot section of wall. (d) Answer part (b) for a 2 -foot section of wall. Round \(\lambda\) to the nearest tenth. (e) Consider your answers to parts (b), (c), and (d). If you had three hairline cracks evenly spread out over a 50 -foot section of wall, should this be cause for concern? The probability is low. Could this mean that you are lucky to have so few cracks? On a 20-foot section of wall [part (c)], the probability of three cracks is higher. Does this mean that this distribution of cracks is closer to what we should expect? For part (d), the probability is very small. Could this mean you are not so lucky and have something to worry about? Explain your answers.

Consider a binomial experiment with \(n=7\) trials where the probability of success on a single trial is \(p=0.30\) (a) Find \(P(r=0)\) (b) Find \(P(r \geq 1)\) by using the complement rule.

Chances: Risk and Odds in Everyday Life, by James Burke, reports that only \(2 \%\) of all local franchises are business failures. A Colorado Springs shopping complex has 137 franchises (restaurants, print shops, convenience stores, hair salons, etc.). (a) Let \(r\) be the number of these franchises that are business failures. Explain why a Poisson approximation to the binomial would be appropriate for the random variable \(r\) What is \(n\) ? What is \(p\) ? What is \(\lambda\) (rounded to the nearest tenth)? (b) What is the probability that none of the franchises will be a business failure? (c) What is the probability that two or more franchises will be business failures? (d) What is the probability that four or more franchises will be business failures?

The owners of a motel in Florida have noticed that in the long run, about \(40 \%\) of the people who stop and inquire about a room for the night actually rent a room. (a) How many inquiries must the owner answer to be \(99 \%\) sure of renting at least one room? (b) If 25 separate inquiries are made about rooms, what is the expected number of inquiries that will result in room rentals?
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