4. Probability

According to the U.S. Department of Education, 42.8% of 3-year-olds are enrolled in day care. What is the probability that a randomly selected 3-year-old is enrolled in day care?

[NW] Home Runs The Wall Street Journal regularly publishes an article entitled “The Count.” In one article, The Count looked at 1000 randomly selected home runs in Major League Baseball.

b. Of the 1000 home runs, it was found that 296 were dropped when a fan had a legitimate play on the ball. What is the probability that a randomly selected home run is dropped?

Police Complaints The Chicago Tribune analyzed 17,713 complaints by citizens against Chicago police officers.

a. Of the 17,713 complaints against police officers, it was found that 7296 were accompanied by a signed affidavit, which is required by state law for the complaint against the officer to proceed. What is the probability that a randomly selected complaint is accompanied by a signed affidavit?

Favorite Day to Order Takeout A survey was conducted by Wakefield Research in which participants were asked to disclose their favorite night to order takeout for dinner. The following data are based on their results.

b. What is the probability a randomly selected individual would choose Friday as their favorite night to order takeout?

"[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.

b. What proportion of the days were you stuck by a red light after 15 days?"

"[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.

f. What is the estimate of the probability of the light being red when you reach the intersection?"

[NW] Medicare Fines In an effort to reduce the number of hospital-acquired conditions (such as infection resulting from the hospital stay), Medicare officials score hospitals on a 10-point scale with a lower score representing a better patient track record. The federal government reduces Medicare payments to those hospitals with the worst scores. The following data represent the scores received by Illinois hospitals.

c. Determine the probability that a randomly selected hospital in Illinois has a score less than 9.

Cigar Smoking The data in the following table show the association between cigar smoking and death from cancer for 139,704 men. Note: Current cigar smoker means cigar smoker at time of death.

b) If an individual is randomly selected from this study, what is the probability that he was a current cigar smoker?

The Placebo Effect A company is testing a new medicine for migraine headaches. In the study, 150 women were given the new medicine and 100 women were given a placebo. Each participant was directed to take the medicine when the first symptoms of a migraine occurred and then to record whether the headache went away within 45 minutes or lingered. The results are recorded in the following table:

(b) If a study participant is selected at random, what is the probability her headache went away within 45 minutes?

Soccer? In a survey of 500 randomly selected Americans, it was determined that 22 play soccer. What is the probability that a randomly selected American plays soccer?

Apartment Vacancy A real estate agent conducted a survey of 200 landlords and asked how long their apartments remained vacant before a tenant was found. The results of the survey are shown in the table. The data are based on information obtained from the U.S. Census Bureau.

c. Find the probability that a randomly selected apartment is vacant for 1–4 months.

Titanic Survivors The following data represent the survival data for the ill-fated Titanic voyage by gender. The males are adult males and the females are adult females.

a. What is the probability that the passenger survived?

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.