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Chapter 5: Problem 1

For a binomial experiment, what probability distribution is used to find the probability that the first success will occur on a specified trial?

Short Answer

Expert verified
Use the Geometric Distribution to find the probability of the first success on a specified trial.

Step by step solution

01

Recognize the Type of Experiment

In this exercise, we need to determine the probability of the first success occurring on a specified trial in a series of independent and identically distributed Bernoulli trials. Since the first success timing is of interest, it suggests a particular probability distribution.
02

Identify the Appropriate Distribution

For an experiment where you determine the probability of the first success on a specific trial, the Geometric Distribution is the appropriate model. This distribution models the number of trials until the first success in a sequence of independent Bernoulli trials.
03

Recall the Geometric Distribution Formula

The probability that the first success occurs on the k-th trial in a geometric distribution is given by the formula: \[ P(X = k) = (1-p)^{k-1} imes p \]where \(p\) is the probability of success on each trial, and \(k\) is the trial number on which the first success happens.

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Key Concepts

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

Binomial Experiment
In statistics, a binomial experiment is a fundamental concept for understanding trial-based probability scenarios. It involves a fixed number of trials, each with two possible outcomes: success or failure. In these experiments, the probability of success remains constant throughout all trials. Key characteristics of a binomial experiment include:
  • Fixed Number of Trials: The experiment is conducted in a predetermined number of trials, denoted as \(n\).
  • Independent Trials: Each trial outcome is independent, meaning the result of one trial does not impact the others.
  • Two Possible Outcomes: Outcomes are classified into success and failure.
In a binomial experiment, we are often interested in finding the probability of achieving a certain number of successes. Although similar, the geometric distribution, which models when the first success occurs, is distinct from the binomial distribution.
Bernoulli Trials
Bernoulli trials are a series of experiments where each trial has exactly two possible outcomes. These are labeled as "success" and "failure." Each trial in the sequence is independent, and the probability of success, labeled as \(p\), remains constant.Some key attributes of Bernoulli trials include:
  • Independent Trials: Each trial is independent, ensuring that the result of one trial does not influence another.
  • Constant Probability: The probability of success (\(p\)) in each trial is the same.
  • Binary Outcomes: Each trial results in either a success or a failure, represented numerically by 1 (success) and 0 (failure).
These trials are the building blocks for more complex probability distributions like the binomial and geometric distributions. In practical terms, real-world scenarios like coin flips or yes-no survey responses are classic examples of Bernoulli trials.
Probability of Success
The probability of success, often denoted by \(p\), is a critical component of various probability models, including binomial and geometric distributions. When dealing with repeated trials like in a geometric distribution, \(p\) represents the chance of achieving a 'success' in any single trial.A few important points about the probability of success include:
  • Consistency Across Trials: In both binomial and geometric experiments, the probability of success remains consistent across all trials. This is a crucial assumption in modeling these scenarios.
  • Playing a Role in Calculations: For a geometric sequence, the formula \(P(X = k) = (1-p)^{k-1} \times p\) utilizes \(p\) to calculate the probability that the first success occurs on the \(k\)-th trial.
  • Influencing Outcome Predictions: The value of \(p\) directly influences the central probabilities and expectations of the distribution, projecting the likelihood and frequency of various occurrences.
Understanding and correctly determining \(p\) is essential, as errors in estimation can significantly affect predictions and interpretations of data in these probabilistic models.

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Most popular questions from this chapter

In his doctoral thesis, L. A. Beckel (University of Minnesota, 1982 ) studied the social behavior of river otters during the mating season. An important role in the bonding process of river otters is very short periods of social grooming. After extensive observations, Dr. Beckel found that one group of river otters under study had a frequency of initiating grooming of approximately \(1.7\) for each 10 minutes. Suppose that you are observing river otters for 30 minutes. Let \(r=0,1,2, \ldots\) be a random variable that represents the number of times (in a 30-minute interval) one otter initiates social grooming of another. (a) Explain why the Poisson distribution would be a good choice for the probability distribution of \(r\). What is \(\lambda\) ? Write out the formula for the probability distribution of the random variable \(x\) (b) Find the probabilities that in your 30 minutes of observation, one otter will initiate social grooming four times, five times, and six times. (c) Find the probability that one otter will initiate social grooming four or more times during the 30-minute observation period. (d) Find the probability that one otter will initiate social grooming less than four times during the 30-minute observation period.

Officers Killed Chances: Risk and Odds in Everyday Life, by James Burke, reports that the probability a police officer will be killed in the line of duty is \(0.5 \%\) (or less). (a) In a police precinct with 175 officers, let \(r=\) number of police officers killed in the line of duty. Explain why the Poisson approximation to the binomial would be a good choice for the random variable \(r .\) What is \(n ?\) What is \(p ?\) What is \(\lambda\) to the nearest tenth? (b) What is the probability that no officer in this precinct will be killed in the line of duty? (c) What is the probability that one or more officers in this precinct will be killed in the line of duty? (d) What is the probability that two or more officers in this precinct will be killed in the line of duty?

Consider two binomial distributions, with \(n\) trials each. The first distribution has a higher probability of success on each trial than the second. How does the expected value of the first distribution compare to that of the second?

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) Quota Problem 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?

What does the expected value of a binomial distribution with \(n\) trials tell you?
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