91Ó°ÊÓ

For each situation, identify the sample size \(n\), the probability of a success \(p\), and the number of success \(x\). When asked for the probability, state the answer in the form \(b(n, p, x)\). There is no need to give the numerical value of the probability. Assume the conditions for a binomial experiment are satisfied. A 2017 Gallup poll found that \(53 \%\) of college students were very confident that their major will lead to a good job. a. If 20 college students are chosen at random, what's the probability that 12 of them were very confident their major would lead to a good job? b. If 20 college students are chosen at random, what's the probability that 10 of them are not confident that their major would lead to a good job? a. If 20 college students are chosen at random, what's the probability that 12 of them were very confident their major would lead to a good job? b. If 20 college students are chosen at random, what's the probability that 10 of them are not confident that their major would lead to a good job?

Step by step solution

01

Identify Parameters for Question a

From the problem statement, the probability of success in a single trial (i.e., a college student is confident their major will lead to a good job) is \(p = 0.53\). The number of trials or sample size is 20 students being chosen at random (\(n = 20\)). The number of successes we are interested in (i.e., students being confident their major will lead to a good job) is 12 (\(x = 12\)).

02

Express the Probability for Question a

The binomial probability \(b(n, p, x)\) is the probability of \(x\) successes in \(n\) trials for event with probability \(p\). So for this case, the answer is \(b(20, 0.53, 12)\).

03

Identify Parameters for Question b

The probability of success in a single trial for this part is a student is not confident their major will lead to a good job. Since there's a 53% chance a student is confident, thus \(p = 1 - 0.53 = 0.47\) for a student to be not confident. Again, the number of trials is 20 students being chosen at random (\(n = 20\)), and the number of successes we are interested in (i.e., students being not confident their major will lead to a good job) is 10 (\(x = 10\)).

04

Express the Probability for Question b

Following the same logic as before, our binomial probability for the second question becomes \(b(20, 0.47, 10)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Sample Size

Understanding sample size is critical when analyzing binomial probability. The sample size, denoted as n, refers to the total number of trials or subjects being considered in an experiment or a survey.

For instance, in the given exercise, when addressing the confidence of college students in their major leading to a good job, the sample size is defined as 20 students. This means that 20 different occurrences or trials are examined, each being an opportunity for the event (student confident in their major leading to a good job) to succeed or fail.

The importance of sample size cannot be overstated as it not only determines the scale of the study but also impacts the reliability of the results. A larger sample size typically leads to more precise estimations of the probability of success, as it tends to represent the population more accurately. However, it is crucial to ensure that the sample is random and representative to avoid bias.

Probability of Success

The probability of success, often symbolized as p, is a key element when discussing binomial distributions. It represents the chance of a single trial resulting in a success.

In our example about college students' confidence in their major, the probability of success for one student being confident is 53%, or 0.53 when expressed as a decimal. Conversely, the probability of a student not being confident can be deduced by taking one minus the probability of confidence, leading to a probability of 47%, or 0.47.

Understanding this probability is vital, as it remains consistent across each trial and is the foundation upon which the binomial distribution is based. When multiple probabilities are involved, such as the probability of confidence and the probability of not being confident, it is essential to define each clearly to avoid confusion while calculating binomial probabilities.

Binomial Distribution

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, with only two possible outcomes: success or failure. This distribution is governed by two parameters: the number of trials n, and the probability of success in each trial p.

Returning to our problem, considering 20 college students and looking to determine how many are confident or not confident about their major leading to a good job, we illustrate the binomial distribution with two key scenarios corresponding to both levels of confidence. The expressions b(20, 0.53, 12) and b(20, 0.47, 10) showcase the potential outcome scenarios.

To calculate the exact probabilities, one would use the binomial probability formula, which incorporates factorial calculations and powers of probabilities. The expression b(n, p, x) serves as a shorthand reference to represent the probability without delving into intricate calculations, making it easier to communicate results concisely.