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Chapter 6: Problem 71

determine whether the given random variable has a binomial distribution. Justify your answer. Lefties Exactly \(10 \%\) of the students in a school are left-handed. Select students at random from the school, one at a time, until you find one who is left-handed. Let \(V=\) the number of students chosen.

Short Answer

Expert verified
The random variable does not have a binomial distribution; it follows a geometric distribution due to the lack of a fixed number of trials.

Step by step solution

01

Identify Binomial Distribution Criteria

A binomial distribution is characterized by the following criteria: (1) There are a fixed number of trials, (2) Each trial is independent, (3) Each trial has only two possible outcomes: success or failure, and (4) The probability of success is the same for each trial.
02

Check Number of Trials

In the given problem, students are selected one at a time until a left-handed student is found. There is no fixed number of trials because the process continues indefinitely until a success is found. This does not meet the fixed number of trials requirement of a binomial distribution.
03

Examine Independence of Trials

Each time a student is randomly selected from the whole school, it is independent of other selections, assuming that the proportion of left-handed students remains the same regardless of previous selections.
04

Determine Trial Outcomes

Each selection of a student can have one of two outcomes: left-handed (success) or right-handed (failure), supporting the binomial distribution requirement of two possible outcomes.
05

Evaluate Constant Probability

The probability that a selected student is left-handed is constant at exactly 10% (or 0.10) for each trial, assuming that the student's handedness does not affect the school population dynamically.
06

Conclude Distribution Type

Despite having independent trials, two outcomes per trial, and constant probability, the lack of a fixed number of trials means this distribution does not meet the criteria for a binomial distribution. This is indicative of a geometric distribution, where trials occur until the first success.

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Key Concepts

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

Random Variable
A random variable is a fundamental concept in probability and statistics. It represents a numerical outcome of a random process. In this exercise, the random variable is defined as \( V \), which is the number of students selected until a left-handed student is found.
This random variable can take different values depending on how many students you need to sample to find a left-handed one.
  • If the first student sampled is left-handed, \( V \) equals 1.
  • If the first two are right-handed and the third is left-handed, \( V \) equals 3.
Here, \( V \) can potentially be any positive integer, depending on how the sampling turns out. Understanding \( V \) through this lens allows you to appreciate its variability and how it captures the essence of the random process involved.
Geometric Distribution
The geometric distribution comes into play when we are dealing with independent trials that continue until the first success occurs. Unlike a binomial distribution, which requires a fixed number of trials, a geometric distribution is defined by
  • A sequence of independent trials
  • Each having the same probability of success
  • Counting how many trials it takes to achieve the first success
In the context of this exercise, selecting students until a left-handed student is found perfectly models a geometric distribution. Here, we don't predefine the number of students to select, which aligns with the hallmark feature of a geometric distribution: an indefinite sequence of trials until success.
Probability of Success
The probability of success in any statistical distribution defines how likely it is for the desired outcome to occur in a single trial. For this exercise, success is defined as choosing a left-handed student, which has a probability of 10%, or 0.10.
This probability remains constant across all trials, which supports the criteria for both binomial and geometric distributions.
  • If selecting another student, the chance remains consistent at 10% that they will be left-handed.
  • This constant probability is foundational for the calculations involved in modeling both binomial and geometric processes.
Understanding this not only helps identify correct statistical models but also provides clarity on how probability influences predictions and outcomes.
Independent Trials
When talking about independent trials, we refer to experiments or selections where the outcome of one does not affect the outcome of others. In probabilistic models, this is crucial for ensuring that past results don't skew future predictions.
For the task at hand, each selection of a student is independent because picking a left or right-handed student doesn't influence the proportion of left-handed students in the school.
  • Each student's selection remains a standalone event, maintaining the 10% chance for left-handedness
  • Independence is a fundamental pillar in defining both binomial and geometric distributions
This core concept allows analyses to yield consistent probabilistic predictions across iterations, essential for accurate statistical modeling.

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Most popular questions from this chapter

Each entry in a table of random digits like Table D has probability 0.1 of being a \(0,\) and the digits are independent of one another. If many lines of 40 random digits are selected, the mean and standard deviation of the number of 0 s will be approximately (a) mean \(=0.1,\) standard deviation \(=0.05\). (b) mean \(=0.1,\) standard deviation \(=0.1\) (c) mean \(=4,\) standard deviation \(=0.05\). (d) mean \(=4,\) standard deviation \(=1.90\). (e) mean \(=4,\) standard deviation \(=3.60\).

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