In Exercises 9–20, use the data in the following table, which lists survey results from high school drivers at least 16 years of age (based on data from “Texting While Driving and Other Risky Motor Vehicle Behaviors Among U.S. High School Students,” by O’Malley, Shults, and Eaton, Pediatrics, Vol. 131, No. 6). Assume that subjects are randomly selected from those included in the table. Hint: Be very careful to read the question correctly.

Texting While Driving If two of the high school drivers are randomly selected, find the probability that they both texted while driving.
a. Assume that the selections are made with replacement. Are the events independent?

Verified step by step guidance

1. Step 1: Calculate the total number of high school drivers surveyed. Add all the values in the table: 731 + 3054 + 156 + 4564.
2. Step 2: Determine the total number of drivers who texted while driving. Add the values in the 'Texted While Driving' row: 731 + 3054.
3. Step 3: Calculate the probability of selecting one driver who texted while driving. Divide the total number of drivers who texted while driving by the total number of drivers surveyed. Use the formula: P(Texted While Driving) = (Number of drivers who texted while driving) / (Total number of drivers).
4. Step 4: Since the selections are made with replacement, the probability of selecting two drivers who both texted while driving is the product of the individual probabilities. Use the formula: P(Both Texted While Driving) = P(Texted While Driving) × P(Texted While Driving).
5. Step 5: Discuss independence. Events are independent if the outcome of one event does not affect the outcome of the other. Since the selections are made with replacement, the events are independent because the first selection does not influence the second.

Key Concepts

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it involves calculating the chance that both randomly selected high school drivers texted while driving. This requires understanding how to compute probabilities based on the total number of favorable outcomes over the total number of possible outcomes.

Independent Events

Two events are considered independent if the occurrence of one does not affect the occurrence of the other. In this scenario, if the selections are made with replacement, the probability of the second driver texting while driving remains unchanged regardless of the first driver's behavior. Understanding independence is crucial for correctly calculating the combined probability of both events.

Sampling with Replacement

Sampling with replacement means that after selecting an individual from a population, that individual is returned to the population before the next selection. This method ensures that each selection is made from the same total population, maintaining the same probabilities for each draw. It is important to recognize this concept when determining the probabilities of independent events in the given problem.