4. Probability

ChatGPT Use According to Pew Research, 26% of U.S. teens ages 13 to 17 using ChatGPT for their schoolwork.

a. What is the probability that four randomly selected 13- to 17-year-olds all use ChatGPT for their schoolwork?

"[NW] Going to Disney World John, Roberto, Clarice, Dominique, and Marco work for a publishing company. The company wants to send two employees to a statistics conference in Orlando. To be fair, the company decides that the two individuals who get to attend will have their names randomly drawn from a hat.

d. What is the probability that John stays home?"

In Exercises 6–10, use the following results from tests of an experiment to test the effectiveness of an experimental vaccine for children (based on data from USA Today). Express all probabilities in decimal form.

If 1 of the 1602 subjects is randomly selected, find the probability of getting 1 who had the vaccine treatment and developed flu.

Putting It Together: Drug Side Effects In placebo-controlled clinical trials for the drug Viagra, 734 subjects received Viagra and 725 subjects received a placebo (subjects did not know which treatment they received). The table below summarizes reports of various side effects that were reported.

d. What is the estimated probability that a subject receiving a placebo would report flushing as an adverse effect? Would this be unusual?

Cardiac Arrest Researchers conducted a prospective cohort study in which male patients who had an out-of-hospital cardiac arrest were submitted to therapeutic hypothermia (intravenous infusion of cold saline followed by surface cooling with the goal of maintaining body temperature of 33 degrees Celsius for 24 hours. Note that normal body temperature is 37 degrees Celsius). The survival status, length of stay in the intensive care unit (ICU), and time spent on a ventilator were measured. Each of these variables was compared to a historical cohort of patients who were treated prior to the availability of therapeutic hypothermia. Of the 52 hypothermia patients, 37 survived; of the 74 patients in the control group, 43 survived. The median length of stay among survivors for the hypothermia patients was 14 days versus 21 days for the control group. The time on the ventilator among survivors for the hypothermia group was 219 hours versus 328 hours for the control group.

f. Based on the results of this study, what is the probability a randomly selected male who has an out-of-hospital cardiac arrest and submits to therapeutic hypothermia will survive? What about those who do not submit to therapeutic hypothermia?

In the game of roulette, a wheel consists of 38 slots numbered 0, 00, 1, 2, ..., 36. (See the photo.) To play the game, a metal ball is spun around the wheel and is allowed to fall into one of the numbered slots.

b. Determine the probability that the metal ball falls into the slot marked eight. Interpret this probability.

College Survey In a national survey conducted by the Centers for Disease Control to determine college students’ health-risk behaviors, college students were asked, “How often do you wear a seatbelt when driving a car?” The frequencies appear in the following table:

a. Construct a probability model for seatbelt use by a driver.

[DATA] Walt Disney Stock The table shows the movement of Walt Disney stock for 30 randomly selected trading days. “Up” means the stock price increased in value for the day. “Down” means the stock price decreased in value for the day, and “No Change” means the stock price closed at the same price it closed for the previous day.

a. Construct a probability model for stock movement of Walt Disney stock.

[DATA] Walt Disney Stock The table shows the movement of Walt Disney stock for 30 randomly selected trading days. “Up” means the stock price increased in value for the day. “Down” means the stock price decreased in value for the day, and “No Change” means the stock price closed at the same price it closed for the previous day.

c. What is the probability that Walt Disney stock is up for a randomly selected day?

[DATA] Walt Disney Stock The table shows the movement of Walt Disney stock for 30 randomly selected trading days. “Up” means the stock price increased in value for the day. “Down” means the stock price decreased in value for the day, and “No Change” means the stock price closed at the same price it closed for the previous day.

e. Would the estimate of the probability of Walt Disney stock price movement improve if we considered 60 randomly selected days instead? Explain.

Merit Badge Requirements To complete the Citizenship in the World merit badge, one must select two of the following seven organizations and describe their role in the world. Source: Boy Scouts of America

1. The United Nations

2. The World Court

3. World Organization of the Scout Movement

4. The World Health Organization

5. Amnesty International

6. The International Committee of the Red Cross

7. CARE

Comment on the likelihood that the pair “The United Nations” and “Amnesty International” will be selected.

Define each of the following.

a. Probability

[DATA] A Random Process: Trains Your daily commute to work requires that you cross railroad tracks. At this particular railroad crossing the trains tend to be long and slow. So, getting stopped by a train will likely make you late for work. You start recording data to determine the likelihood of arriving at the tracks while a train is there. Open data set 5_1_37 at www.pearsonhighered.com/sullivanstats, which contains the day number and whether a train was present, or not, for 200 consecutive days in which you drove to work. The column “Train” shows a series of 0s and 1s. In that column, a 0 indicates there was no train present and a 1 indicates that a train was present. The column “Aggregate Train” represents the cumulative number of times a train was present. The column “Aggregate Proportion Train” represents the cumulative proportion of times a train was present.

d. Were you stopped by a train on the 30th day?

[DATA] A Random Process: Trains Your daily commute to work requires that you cross railroad tracks. At this particular railroad crossing the trains tend to be long and slow. So, getting stopped by a train will likely make you late for work. You start recording data to determine the likelihood of arriving at the tracks while a train is there. Open data set 5_1_37 at www.pearsonhighered.com/sullivanstats, which contains the day number and whether a train was present, or not, for 200 consecutive days in which you drove to work. The column “Train” shows a series of 0s and 1s. In that column, a 0 indicates there was no train present and a 1 indicates that a train was present. The column “Aggregate Train” represents the cumulative number of times a train was present. The column “Aggregate Proportion Train” represents the cumulative proportion of times a train was present.

f. What is the estimate of the probability of being stuck by a train during your commute?  

[DATA] A Random Process: Green Lights On your drives to school each day you feel like there is a light that is always red when you reach it. You decide to record data to determine the likelihood of arriving at the light while it is red. Open the data set 5_1_38 at www.pearsonhighered.com/sullivanstats, which contains the day number and whether the light was red (1), or not (0), for 120 consecutive days in which you drove to school.

d. Were you stuck by a red light on the 50th day?