Binomial Distribution: 동영상 및 연습문제

During a study on $10$ students' habits, the number of students who study at least $2$ hours daily is recorded. Determine if $8$ students studying at least $2$ hours daily is significantly high using the range rule of thumb. Assume the probability of a student studying at least $2$ hours daily is $0.5$.

In a poll about daily reading habits, groups of four individuals were asked if they read books for at least $30$ minutes a day. The random variable $x$ denotes the number of individuals in a group who say they read books for at least $30$ minutes a day. Calculate the probability of exactly one individual in a group saying they read books for at least $30$ minutes a day. Assume the probability that an individual reads books for at least $30$ minutes a day is $p=0.5$.

A science quiz contains three multiple-choice questions, each with four possible answers (a, b, c, d), only one of which is correct. Suppose you randomly guess the answers to the first three questions.

Beginning with WWC (two wrong answers followed by one correct answer), determine the number of different possible arrangements of two wrong answers (W) and one correct answer (C). Then, calculate the probability for each arrangement.

A history quiz contains three multiple-choice questions, each with four possible answer choices (a, b, c, d), only one of which is correct. Suppose a student randomly guesses on all three questions.

Beginning with WWC (two wrong answers followed by one correct answer), make a list of different possible arrangements of two wrong answers (W) and one correct answer (C). What is the probability of getting exactly one correct answer when three guesses are made?

A survey report suggests that $30\%$ of adults prefer reading e-books over printed books. Assume that $10$ adults are randomly selected. Find the probability that exactly $7$ of the selected adults prefer reading e-books.

According to a recent survey, $25\%$ of people prefer to work remotely full-time. Assume that $9$ people are randomly selected. Find the probability that at least $6$ of the selected people prefer to work remotely full-time.

A technology survey found that $35\%$ of smartphone users use voice assistants daily. Assume that $8$ smartphone users are randomly selected. Find the probability that fewer than $3$ of the selected users use voice assistants daily.

According to a recent survey, $45\%$ of college students prefer online classes over traditional in-person classes. Assume that $8$ college students are randomly selected. Find the probability that no more than $3$ of the selected students prefer online classes.

In a recent study on workplace diversity, a company reports that $35\%$ of its employees are from minority groups. A hiring committee randomly selects $60$ applicants to form a short list, but only $6$ of them are from minority backgrounds. Find the probability of selecting $6$ or fewer minority applicants out of $60$, assuming the selection was truly random. Is the result of $6$ minorities significantly low? What might this suggest about the selection process?

A study shows that $5\%$ of emails sent by a company are not delivered successfully. If the company sends out $20$ emails, what is the probability that exactly $4$ emails will not be delivered successfully?

In a study, it was observed that $2.8\%$ of online transactions were fraudulent. If $20$ transactions are randomly selected, what is the probability that at least one transaction is fraudulent?

Based on a recent technology survey, $42\%$ of smartphone users enable two-factor authentication (2FA) for added security. For each of the following, assume that $15$ smartphone users are randomly selected. What is the probability that exactly $10$ of the selected users have enabled two-factor authentication?

In a certain city, it is found that $5\%$ of the drivers run red lights. If a traffic study observes $30$ drivers at an intersection, find the mean and standard deviation for the number of drivers expected to run red lights.

According to a health survey, $2\%$ of a population is known to have a certain genetic disorder. If a doctor tests $12$ individuals from this population, what is the probability that none of them has the disorder?

During a promotional event, a company decides to give away free samples of their new product to the first $100$ visitors. Surprisingly, $98$ of the first $100$ visitors were given a variant A of the product, while only $2$ received variant B. If the distribution was supposed to be random with an equal chance for each visitor to receive either variant, what is the probability of $98$ or more visitors given a variant A sample?

In a school science fair, $30$ projects are judged, and the top $3$ projects receive awards. Although the selection process is designed to be random, it has been observed that environmental science projects have won the top $3$ awards $28$ times out of $30$ fairs. Assuming that environmental science projects and other types of projects have an equal probability of winning, calculate the probability of observing $28$ or more such outcomes.

A pharmaceutical company tested a new drug on $1000$ patients and found that $700$ of them showed improvement. Assuming the expected success rate of the drug is $70\%$, use the range rule of thumb to identify the limits for significantly low and high improvement rates. Is the observed improvement rate of $700$ patients significantly low or high?

In a study on the germination of $800$ seeds, it was observed that $240$ seeds germinated successfully. The hypothesis was that $30\%$ of the seeds would germinate under the given conditions. Find the probability of exactly $240$ seeds germinating successfully.

In a study on the germination of $800$ seeds, it was observed that $240$ seeds germinated successfully. The hypothesis was that $30\%$ of the seeds would germinate under the given conditions. Find the probability of $240$ or more seeds germinating successfully.

In a study about adopting electric vehicles, $620$ out of $1200$ participants (or $51.67\%$) agreed that incentives for electric vehicle purchases should be increased. Assuming the true agreement rate is $50\%$, is the result of $620$ significantly high? What does this suggest about the public opinion on increasing incentives for electric vehicle purchases?

A local health department finds that $3\%$ of a community is allergic to a common vaccine. Assume that a group of $12$ individuals is randomly selected for vaccination. If $2$ out of the $12$ individuals experience an allergic reaction, is this result considered significantly high?

In a study of a group of five employees about working on weekends, the probabilities for employees agreeing to work were noted. In the table provided, the random discrete variable $x$ represents the number of employees in a group who agree to work on weekends.

Determine whether $2$ is a significantly low number of employees who agree to work on weekends. Use the range rule of thumb.

In a study of a group of five employees about working on weekends, the probabilities for employees agreeing to work were noted. In the table provided, the random discrete variable $x$ represents the number of employees in a group who agree to work on weekends.

Find the probability of having $3$ or more employees who agree to work on weekends.

In a study of a group of five employees about working on weekends, the probabilities for employees agreeing to work were noted. In the table provided, the random discrete variable $x$ represents the number of employees in a group who agree to work on weekends.

Is $3$ a significantly high number of employees who agree to work on weekends?

In a study about pet ownership, $300$ people were asked if they own a dog. The results are summarized below. Determine if this summary can be considered a probability distribution. Justify.

In a local school raffle, $5000$ tickets are sold for $\text{\$}2$ each. Five winning tickets will be drawn at random. If you purchase one ticket, what is the probability that your ticket is one of the winners?

A study found that $41\%$ of households regularly separate their recyclable materials. Suppose $15$ households are randomly chosen. If $11$ of these households regularly recycle, is $11$ considered significantly high? Why or why not?

In a weekly lottery, you pick two digits (from $00$ to $99$) with repetition allowed. Each week, one winning number is drawn at random. If you buy one ticket every week for a year consisting of $52$ weeks, find the mean number of wins you can expect.

A consumer study found that $41\%$ of shoppers prefer buying organic groceries. Suppose $15$ shoppers are randomly selected. What is the probability that at least one of the selected shoppers prefers organic groceries?

A survey reports that $20\%$ of people who regret buying a certain car model say they made the purchase too late in the model's lifecycle. Suppose five such individuals are randomly selected.

What is the probability that none of the selected individuals say they bought the car too late?