Do these situations involve Bernoulli trials? Explain. a)...

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.

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.

Key Concepts

Sampling Without Replacement (Hypergeometric Distribution)

When sampling from a finite population without replacement, the outcomes of the trials become dependent because the composition of the population changes with each draw. This situation is better modeled by the hypergeometric distribution rather than the binomial distribution, which assumes independent trials with constant probabilities.

Constant Probability of Success

A core requirement for modeling Bernoulli trials is that the probability of success remains the same for every trial. If the likelihood of success changes from trial to trial—due to factors like sampling without replacement or varying conditions—the process no longer fits the Bernoulli model, and alternative probability models may be more appropriate.

Independence of Trials

Independence in the context of Bernoulli trials indicates that the outcome of any individual trial does not affect the outcomes of the others. This property is critical because any dependency between trials can violate the assumptions behind many probability models, including the binomial distribution.

Bernoulli Trials

Bernoulli trials refer to experiments or processes that have exactly two possible outcomes – often labeled as 'success' and 'failure'. Each trial is independent, and the probability of success remains constant across all trials. This concept is central when determining whether a series of events or experiments can be modeled using a binary framework.

Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is used when the total number of trials is predetermined and each trial’s outcome is binary, making it a natural model for assessing probabilities in repeated, identical trials.

Transcript

00:01 We're given some more situations, and we need to decide if these also include bernoulli trials.

00:06 Now, the textbook defines our requirements for bernoulli trials as one.

00:10 You could only have two possible outcomes, where one is a success and the other is a failure.

00:16 Two, the probability of that success needs to stay constant, and three, the trials need to be independent.

00:23 You also have the 10 % condition, which applies to when you're sampling without replacement.

00:29 So anytime you're taking things from the population, that will obviously change the probability of success.

00:35 However, you can still continue as long as the sample size that you are taking is less than 10 % the size of the population.

00:44 So for our first scenario, we are rolling five dice and need to get at least two sixes to win the game.

00:51 Now, your first thought might be that a die has six sides and six options, so this isn't a bernoulli trial.

00:57 However, we have defined that sixes are a success and anything else is a failure.

01:04 So we actually do have two options, six and not six.

01:09 So yes, this is a renewal trial.

01:12 Our second situation, we record the distribution of eye colors found in a group of 500 people.

01:18 Obviously, we have several different types of eye colors more than two.

01:23 And in this scenario, we haven't defined that any one eye color is considered a success.

01:28 So no, that is not a brunali trial.

01:33 A manufacturer recalls a doll because about 3 % have buttons that are not properly attached.

01:39 Customers return 37 of these dolls to the local tory store.

01:42 Is the manufacturer likely to find any dangerous buttons? well, let's go through the requirements.

01:48 One, are there only two outcomes? yes.

01:51 Are there a button that is or is not properly attached? two, does the probability of success stay constant? it says we have a 3 %? a chance of a button not being properly attached...

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