Question

Do these situations involve Bernoulli trials? Explain. a) You are rolling 5 dice and need to get at least two 6 's to win the game. b) We record the distribution of eye colors found in a group of 500 people. c) A manufacturer recalls a doll because about \(3 \%\) have buttons that are not properly attached. Customers return 37 of these dolls to the local toy store. Is the manufacturer likely to find any dangerous buttons? d) A city council of 11 Republicans and 8 Democrats picks a committee of 4 at random. What's the probability they choose all Democrats? e) A 2002 Rutgers University study found that \(74 \%\) of high school students have cheated on a test at least once. Your local high school principal conducts a survey in homerooms and gets responses that admit to cheating from 322 of the 481 students.

Short answer

a) Yes, indirectly; b) No; c) Yes; d) No; e) Yes.

Step-by-step solution

Understanding Bernoulli Trials

A Bernoulli trial is a random experiment with exactly two possible outcomes, often termed as "success" and "failure." The probability of success, denoted by \( p \), is the same for every trial, and the trials are independent.

Evaluating Scenario (a)

In scenario (a), rolling 5 dice and hoping to get at least two 6's does not fit a classic Bernoulli trial setup because we are looking for a range of successes (at least 2). Instead, each die roll can be considered as one Bernoulli trial where getting a 6 (success) or not (failure) are the two outcomes. This scenario can be pieced together from multiple Bernoulli trials.

Evaluating Scenario (b)

Scenario (b) involves recording the distribution of eye colors in a group of 500 people. This does not fit the definition of Bernoulli trials as there are more than two outcomes (multiple eye colors possible) and no clear notion of repeatedness or independence characteristic of such trials.

Evaluating Scenario (c)

For scenario (c), considering whether returned dolls have improperly attached buttons resembles a Bernoulli trial. Each doll represents a trial with two outcomes: a button is defective (success) or not (failure), assuming each doll's state is independent and the same probability \(3\%\) applies.

Evaluating Scenario (d)

Scenario (d) involves selecting a committee and checking a group composition (all Democrats). This is not a Bernoulli trial since choosing committee members changes the probability of subsequent selections, violating the condition of independent trials with a constant probability.

Evaluating Scenario (e)

Scenario (e) involves conducting a survey in which each student's response (cheated or not) could be seen as a Bernoulli trial. Each student's response is independent, and the probability \(74\%\) (cheated) vs. \(26\%\) (not cheated), signifies two outcomes with constant probabilities.

Key concepts

Probability Theory

Probability theory is a crucial part of mathematics that deals with the likelihood of events occurring. In the context of Bernoulli trials, it provides the framework to calculate the probability of "success" or "failure" in a series of experiments. For each Bernoulli trial, the probability, often denoted as \( p \), remains constant throughout the trials. This means that each experiment or trial has the same chance of success, no matter the outcome of previous or concurrent trials.

Understanding probability theory helps in making predictions and decisions based on the likelihood of certain outcomes. In many everyday applications, we use probability to assess risks or predict probable results, making it an essential tool in various fields such as finance, science, and social sciences. An essential aspect of probability theory is that it helps in analyzing scenarios where there are two distinct outcomes, identified as success and failure in Bernoulli trials.

Independent Trials

Independent trials are a key concept in the realm of Bernoulli processes. This independence means that the result or outcome of one trial does not affect the outcome of any other trial. For example, when flipping a coin, the result of the first flip does not impact what happens on the second flip — each flip is independent from one another.

In evaluating scenarios for Bernoulli trials, identifying independence is crucial. Consider the example of the doll recall process in the original exercise. Each doll return represents an independent event, since whether one doll has a defective button does not influence the button condition on another doll.

Independence allows us to calculate complex probabilities by multiplying the probabilities of individual trials. If trials were dependent (like drawing cards from a deck without replacement), the probabilities would change as events unfold, making calculations much more intricate.

Binary Outcomes

Binary outcomes signify that each trial in a Bernoulli process must have exactly two possible outcomes, frequently categorized as "success" or "failure." This concept is fundamental for defining and recognizing Bernoulli trials, as seen in many scenarios.

For instance, in the survey conducted in the local high school, each student's response is a binary outcome — they either admit to cheating or they do not. This makes it a suitable example of a Bernoulli trial, with probabilities based on the historical data provided (74% admitted to cheating).

Binary outcomes simplify the process of probability calculations, enabling us to utilize binomial distributions to assess the likelihood of certain numbers of successes occurring. Understanding and identifying binary outcomes is key in problem-solving scenarios involving random experiments with these two possible outcomes.